Dynamics, Statistics and Projective Geometry of Galois Fields V.I. Arnold
2 Abstract. This book represents a 2-hours long conference to the Moscow highschool children at the Moscow State (Lomonosov) University MGU, read there the 13-th November 2004. It described some astonishing recent discoveries of the relations of the Galois fields to dynamical systems ergodic theory, to statistics and chaos, and also to the geometry of the projective structures on finite sets. Most of these recent discoveries summarized empirical studies, and some of the conjectures, provided by these numerical experiences, are still unproved, in spite of their simple statements, quite accesible to the highschool children (who might sudy them empirically, being virtually assisted by the computers). Together with these empirical studies continuation, it would be nice to investigate some remaining theoretical questions, like the natural problem of the projective permutations intrinsic characterization among all the permutations of a finite set: one should understand, which geometrical features of some special permutations of dozen points are making these special permutations projective, distinguishing them from the nonprojective permutations. These researches had been partially supported by RFBR, grant 05-01-00104. Keywords : Mathematics, algebra, geometry, number theory, Galois fields, dynamical systems, ergodic theory, mathematical statistics, projective geometry, projective line, Frobenius transformation, chaoticity, randomness, equipartition, primes, Euler group, Euler function, Cesaro averaging, geometrical progression, Fermat-Euler congruences, little Fermat theorem, binomial and multinomial coefficients, Girard-Newton formula.
Contents 1 What is a Galois field?
5
2 Galois field’s organization and table
13
3 Chaoticity and randomness of Galois field table’s numbers
21
4 Equipartition of geometrical progressions along a finite onedimensional torus 29 5 Adiabatic study of distribution of geometrical progressions of residues 43 6 Projective structures, generated by a Galois field
51
7 Calculation of projective structures of finite projective lines, generated by fields of p2 elements, and of Frobenius transformations group action on these lines and on these structures 63 8 Cubic tables of fields
79
3
4
CONTENTS
Chapter 1 What is a Galois field? A Galois field is a field, having a finite number of elements. Such fields belong to the small quantity of the most fundamental mathematical objects, serving to describe all other mathematical structures and models. The examples of such fundamental objects are the well known prime numbers, p = 2, 3, 5, 7, 11, 13, 17, 19, 23, . . . , 997, 1009, . . . , these are the positive integers, which have each only 2 integer divisors (1 and itself), the number 1 being not prime. The immediate natural science question, to which this notion leads, is already rather difficult: is the set of all the primes finite? (that is, whether the above sequence of primes might be continued indefinitely?) The answer to this question had been discovered several thousands years ago: the prime numbers sequence is infinite, a maximal prime number does not exist. To prove it, it suffices to consider the number (2 · 3 · 5 · · · · · p) + 1 , which provides the residue 1 while we divide it by any prime number 2, 3, . . . , p. This number is not, therefore, divisible by any of them. Hence it has a prime divisor, which is greater, than p. Therefore no maximal prime number p may exist. This remarkable mathematical reasoning is rather avoiding the question of highest interest from the natural sciences view-point: how often are primes encountered in the sequence of all the natural numbers {1, 2, 3, 4, 5, 6, . . . }? 5
6
What is a Galois field?
Are the intervals between the consecutive prime numbers growing (while the numbers we consider become large)? What is the decimals number of the millionth prime? The first natural scientist, studying this problem, has been Adrien Marie Legendre (1752-1833), who had considered (in the XVIII Century) the tables of the primes up to 106 and who had discovered empirically the following density decline of the primes distribution law: the average distance between the consecutive prime numbers, of order of n, grows with n like ln n (the natural logarithm), that is the like logarithm with base e ≈ 2, 71828 . . . , the “Euler number” e being e = lim
k→∞
1 1+ k
k
∞ X 1 . = m! m=0
For instance, ln 10 ≈ 2, 3, and the average distance between the consecutive primes close to 10 is slightly greater than 2, since 7 − 5 = 2 , 11 − 7 = 4 , 13 − 11 = 2 . The primes in the region of n = 100 are 89, 97, 101, 103, their average distance being therefore 4 32 . This distance should be compared to ln 100 = 2 ln 10 ' 4, 6 of the Legendre law, and it is thus confirmed satisfactory already for n = 100. Of course, the existence of the pairs of the twins (that is, of the prime pairs, whose difference is 2, like for 5 and 7, 17 and 19, 29 and 31) contradicts the growing of the distances between the consecutive prime numbers (provided that the twins number is infinite, which is conjecturally true: this infinity is one of the most celebrated unproved conjectures of the modern number theory). Unfortunately, the Legendre empirical observations had not been appreciated by the mathematical community of his time, since “he had proving nothing, considering only some millions of examples”. It is true, that he succeeded to “deduce” from his empirical statistical observations his “law”, being unable to provide the strict mathematical proof of the limit asymptotical coincidence of the averages of the distances with his proposed value ln n for n → ∞. Kolmogorov told me several times on his hydrodynamical turbulence studies: “do not try to find in my works any theorem, proving my statements: I
Chapter 1
7
am unable to deduce these statements from the basic (Navier-Stokes) equations of this theory. My results on the solutions of these equations are not proved, they are true, which is more important, than all proofs”. The first who appreciated the Legendre discoveries had been the Russian mathematician Tchebyshev. He proved first, that even if the averaged distance between the consecutive primes in the neighbourhood of a large number n does not behave asymptotically as ln n, its relation to this Legendre suggestion remains bounded, the average distance belonging to the interval between c1 ln n and c2 ln n (where c1 < c2 were explicitly calculated). Later he proved more: provided that the oscillations between the above limits would extinct while n grows, leading for the average distance to the asymptotics c ln n with some constant c, then the constant c can not be different from 1. This is yet unsufficient to prove the Legendre asymptotical formula, since there remains the possibility of the nonextincting oscillations between c1 ln n and c2 ln n, never leading to the c ln n behavior. However later (about 100 years after the Legendre discovery) two celebrated mathematicians, Hadamard (from France) and de la Vall´ee Poussin (from Belgium) proved, that the oscillations are indeed extincting for n → ∞, providing some c ln n asymptotical behaviour of the averaged distance between the consecutive primes in the neighbourhood of n. The world mathematical community claims therefore, that Hadamard and de la Vall´ee Poussin made a great discovery (of the statistics of the large prime numbers distribution). It seems to me, that this claim is rather unfair. These great mathematicians had only proved the existence of the distribution law (the existence of the constant c, unknown to them). Both facts of the natural science (the asymptotical proportionality of the average distance to ln n and the fact that the proportionality coefficient equals 1) were discovered by Legendre and Tchebyshev, to whom one should attribute the great discovery of the primes density statistics, described above. In this high-school children conference I shall follow rather Legendre, than Hadamard: I shall talk on the empirical numerical observations, suggesting some new (and astonishing) laws of nature, whose transformation to the mathematical theorems state might wait some hundred years (as it had happened to the primes distribution law), in spite of the fact, that the discoveries of these new laws are quite accessible to the schoolchildren (even using no computers, while the computerized experiments might accelerate
8
What is a Galois field?
the empirical experiences1 ). Besides the prime numbers, another example of fundamental mathematical objects is provided by the regular polyhedra (called also “Platonic polyhedra”, since they had been discovered by others). There are five bodies: the tetrahedron (with 4 faces), the octahedron (with 8 faces), the cube (with 6 faces), the icosahedron (from the Greek “icos” for its 20 faces) and the dodecahedron (from the Greek “dodeca” for its 12 faces) – see figure 1.1.
Tetrahedron
Octahedron
Cube
Icosahedron
Dodecahedron
Figure 1.1: Regular polyhedra. The dodecahedron had been used by Kepler to describe the distribution of the planetary orbits radii in the Solar system. The regular polyhedra are strangely related to a domain of Physics which seems to be quite different – to the theory of the optical caustics, which provides, for instance, first – the explanation of the rainbow phenomenon (the rainbow angular radius being α = 42◦ ) and second – the theory of galaxies concentration in the Universe large scale structure. Kolmogorov explained, that the special beauty of the mathematical theories is due to the unexpected relations between quite different natural phenomena (say, between the theories of the electric and magnetic fields, provided by the Maxwell equations), which relations Mathematics discover. Unlike for the fundamental objects of the examples above, the Galois fields applications to the natural sciences are yet to be discovered. I hope, that they will appear rather soon, and I would like to shorten the remaining waiting time by my geometric presentation of the Galois fields theory. My description is closer to the natural science approach, than to the axiomatico1
In my personal experiments, leading me to the results below, no computers had been used, and my students, who had verified that the computerized experiments provide the same answers, discovered that my calculations contained several times less mistakes than that of the computers.
Chapter 1
9
α
α
Figure 1.2: The rainbow origin. algebraic superabstraction style, dominating the existent presentations of this algebraic theory. The simplest example of a Galois field is the residues field modulo a prime number p (figure 1.3). 1 0
2
3
4
Figure 1.3: A finite circle: Galois field Z5 . Thus, for p = 2 we get the field, consisting of two elements: Z2 = {0, 1} , with its usual arithmetics 0+0=0 ,
0+1=1+0=1 ,
0·0= 0·1= 1·0 =0 ,
1+1=0 ,
1·1=1 .
This “binary” arithmetics is the base of the computers, acting in the binary system. Thus, the simplest Galois field is extremely useful: (the field Z2 ) =⇒ (computers) .
10
What is a Galois field?
The general field notion is very similar to the preceeding example: there are two operations (called “addition” and “multiplication”), having the usual properties of commutativity, associativity and verifying the ordinary distributivity law; and one can divide the field’s elements by every element of the field, different from 0. The residues of the division by 3 form the field Z3 , consisting of 3 elements {0, 1, 2} (where 1/2 = 2, since 2 · 2 = 1 for the residues modulo 3: (3a + 2)(3b + 2) = 9ab + 6a + 6b + 4 = 3c + 1). Very differently, the 4 residues for the division of the integers by 4 do not form a field, since the element 2 can not be inverted (the residue 2x is sometimes 0 sometimes 2, but it is different from 1, whatever residue would be x). However, there does exist a field of 4 elements (the operations being different from the residues arithmetics modulo 4). To find these operations is a useful exercise, which is neither too difficult, nor too easy for a beginner. The finite fields are called Galois fields, since he had discovered the following two remarkable properties of these fields: 1. The elements number of a finite field is an integer of the form pn , where p is a prime; and for any prime p and any natural n there exists a finite field having just pn elements. Thus, there exist fields with 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27 elements, but there does not exist any field whose elements number is 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26 . 2. The field, having pn elements, is defined by the number of its elements unambiguously (up to a field’s isomorphism). Thus, a computer using the field Z2 at Moscow, and another computer, working in Paris, might use two different copies of this field. Say, one might denote the Paris field elements α and β (instead of 0 and 1), defining the operations by the table α+α=β+β =β , α·α=α ,
α+β =β+α=α ,
α·β = β ·α =β ·β = β .
Chapter 1
11
But this field is isomorphic to the standard residues field Z2 (differing only in the notations, α ∼ 1 and β ∼ 0). The notations independence of the content of the phenomena is the base of the relativity theory and of the whole relativistic physics. I shall not write here the proofs of the existence and of the uniqueness theorems for the field of pn elements, formulated above. I shall describe instead the operations in this field by explicit tables. Strangely, I had not seen in the published form the natural-science oriented description of the finite fields, presented below. Every field contains the 0 element (zero), which does not change any element, to which it is added. All the other elements of the field form the multiplicative group of the field (being a group for the multiplication operation), since each nonzero element can be inverted. This group is always cyclic: there exists such an element A of the field, that every non-zero element of the field has the form Ak (where 1 ≤ k ≤ z −1 for the field of z = pn elements). I shall not prove the cyclicity theorem (while its proof is not too difficult), since this theorem adds to the theory, described below, only the following axiomophilic addition: in the nature there are no other finite fields, different from the fields with a cyclic multiplicative subgroup. In other words, we might consider the theory, explained below, to describe the finite fields with an additional axiom (that the field’s multiplicative group is cyclic) – the existence of the primitive element A, whose powers provide all the nonzero elements of the field. The constructions and results, explained below, belong to the theory of such fields. The absence of any different finite field is a (nice) addition to this theory, but the theory itself does not depend on this additional property of our axioms. It is interesting to observe, that just the exagereated attention to the difficult studies of the axioms independence makes the algebraic and abstract theories of the mathematicians unnecessarily difficult and hostile for the natural scientists. Thus, the Lobachevsky plane is simply the unit circle interior disc, whose interior points are called “Lobachevsky points”, and whose “Lobachevsky lines” are the unit circle chords. The boundary circle (which does not belong to the Lobachevsky plane) is called “the absolute”. It is very easy to see, that these objects (forming the so-called F. Klein
12
What is a Galois field?
model of the Lobachevsky plane since they had been invented by A. Cayley) verify all the Euclidean geometry axioms (“there exists one and only one line, connecting two given points”, etc.), except the “parallels axiom”: there exist an infinity of Lobachevsky lines going through a given Lobachevsky point and having no common Lobachevsky points with a given Lobachevsky line, disjoint with the given Lobachevsky point (that is, an infinity of chords, see figure 1.4).
Figure 1.4: Lobachevsky plane. These (obvious) natural science facts can be completed by a (difficult) theorem of the axiomophils: there exists no other Lobachevsky plane, different from the Klein model, described above (of course up to isomorphisms: the theorem states, that the Lobachevsky plane axioms imply the isomorphism of this plane to that of the Klein model). It is interesting, that Lobachevsky was unable to prove his main natural science belonging (and quite remarkable) statement: the parallelism axiom of the Euclidean geometry is independent from the other axioms (that is, it can not be deduced from them). The model, described above (and invented many years later, than Lobachevsky worked) proved just this independence statement. Indeed, if the wrongness of the Euclidean parallels axiom would imply a contradiction (which contradiction would be just the axiom’s proof), then the model would be contradictory too, providing therefore contradiction inside the usual Euclidean geometry (concerning the ordinary geometry of the chords of one circle). The proofs of the fundamental mathematical facts are in many cases much simpler, than the axiomophilic details, making so difficult the mathematical textbooks.
Chapter 2 Galois field’s organization and table The multiplication in the Galois field, consisting of n elements, 0 and {Ak }, 1 ≤ k ≤ n − 1 is simply the addition of the “logarithms” k of the elements (considering these logarithms as the residues of the numbers k modulo n−1): 0 · Ak = 0 ,
Ak · A` = Ak+`
(if k + ` > n − 1, one replaces the sum by k + ` − (n − 1) to reduce the sum to a value, smaller than n). It remains to define the addition operation. Denoting the field’s element k A by the sign k, we arrive to the following tropical operation ∗ over these logarithms: Ak + A` = Ak∗` . The modern term “tropical”, meaning “exotic”, is used when one lowers the level of the algebraic operations, transforming the multiplication to the addition, replacing the addition by the lower level “tropical addition” operation, with respect to which the usual addition is as distributive, as is the usual multiplication with respect to the usual addition: x(y + z) = xy + xz ,
x + (y ∗ z) = (x + y) ∗ (x + z) .
An example of such a tropical addition is the operation x ∗ y = max(x, y) for the real numbers. One would be able to obtain this tropical operation from the usual addition using the logarithms accompanied by the short waves 13
14
Galois field’s table.
asymptotical expansion of quantum mechanics, where the wave length h is approaching 0. The relation x ∗h y = ln(ex/h + ey/h ) h defines the tropical addition operation ∗h , tending to max(x, y) for h → 0. While all these things are obvious, they imply a nonobvious “tropical” conclusion: replacing the multiplication and the addition operations with their tropical versions (addition and maximum), one can transform many formulae and theorems of the calculus (like the Fourier series theory) into its (nonevident) “tropical” versions, providing interesting results in the convex calculus and linear programming theories. Consider for simplicity the case of the field F of z = p2 elements. It contains the “scalar” elements 1, 2 = 1 + 1, . . . . The field being finite, one of the sums would coincide with the other. Hence some sum of m ones (equal to the difference of the coincident sums) equals to zero, m = 1 + · · ·+ 1 = 0. We shall suppose the repetitions number m to be the minimal value, providing the 0 scalar. We shall prove now, that m = p. Indeed, call any element x equivalent to any element x + 1 + · · · + 1 (where the number of the ones is at most m). Each equivalence class consists from m elements, and these classes are disjoint. Therefore, the scalar elements number, m, is a divisor of the number p2 of the elements of the field. Thus m is either p or p2 . The second case is impossible. Indeed, consider the scalar element x = 1 + · · · + 1 (p times). This element of the field having p2 elements has no inverse element, since no integer of the form pq provides the residue 1 while one divides it by p2 . Therefore, x = 0, and the scalars number is m = p. Consider the element 1 together with the primitive element A of our field. Adding each of them less than p times, we create the p2 sums uA + v1. All these elements of the field are different (otherwise we would obtain A = (−v/u) · 1, and therefore all the elements of the field would be scalars, which is impossible, since the number of the scalars is p, which is smaller than p2 ). Thus the field of p2 elements consists exactly of the linear combinations F = {uA + v1} with coefficients u ∈ Zp , v ∈ Zp . In this sense we had distributed all the elements of the field in the cases of a p × p square (or rather of the “finite torus” Z2p of figure 2.1, being the 2-plane over the field Zp ).
Chapter 2
15
Therefore, we filled the z = p2 cells of this finite torus by the p2 “logarithmical symbols” {∞; 1, . . . , z − 1} (where the symbol k, which is a residue modulo z − 1, denotes the element Ak of the field F , the symbol ∞ representing1 the field’s zero element). R2
Figure 2.1: The continuous torus and the finite torus, consisting of 4 points. This filling provides the tropical operation ∗ simple description. Namely, the sum of the field’s elements corresponding to the symbols k and `, Ak = u0 A + v 0 1 ,
A` = u00 A + v 00 1 ,
is (u0 + u00 )A + (v 0 + v 00 )1 = Ak∗` . Therefore (figure 2.2), the symbol k ∗ ` fills in the table of the field the vector sum of the places of the symbols k and `: the addition operation of the field F (consisting of p2 elements) is represented by the vector addition of the places of the added elements in the table of the field. u
k∗` k
` v
Figure 2.2: Tropical addition of the numbers k and ` in the table of the field. Thus, to describe the field of z = p2 elements it suffices to calculate the places (uk , vk ) of the elements Ak = uk A + v k 1 1
(1 ≤ k ≤ z − 1)
The schoolchildren suggested at the lecture to denote ln 0 by −∞, but I leave the symbol ∞, being unaware, whether A > 1 in F .
16
Galois field’s table.
in the table of the field. This calculation is an (easy) extension of the Fibonacci numbers ak recurrent construction (these numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . , ak+2 = ak+1 + ak , describe the rabbits population growth). Namely, suppose that in field F we have A2 = αA + β1 ,
(α ∈ Zp , β ∈ Zp ) .
(2.1)
Then we find in F the relation A3 = A(αA + β1) = α(αA + β1) + βA = (α2 + β)A + αβ1 . Continuing in this way, we get the recurrent relation, expressing recurrently the places of the elements Ak in the table of the field, uk+1 = αuk + βvk ,
vk+1 = αvk .
(2.2)
Therefore, the 2 residues α and β (modulo p) provide consecutively the places (uk , vk ) of all the elements Ak in the table of the field. To obtain the table, it remains to choose the values of the parameters α 2 and β. One should choose them in such a way, that, first, Ap −1 = 1, (that is, up2 −1 = 0, vp2 −1 = 1) and, secondly, all the preceding vectors (uk , vk ), (1 ≤ k < p2 − 1) should be different from the vector (0, 1). In principle, one might try in turn all the p2 pairs of residues (α, β) to choose the convinient parameters values. The number of trials is even not too large. For instance, if p = 5 both conditions above are fulfilled by the pair α = β = 2. However, one might seriously accelerate the choice of the parameters values, using the Pascal triangle of the binomial coefficients. Namely, it is easy to prove the following explicit formula for the recurrent relations (2.2) X t uk = αs β t Cs+t , (2.3) where the powers s and t are related by the homogeneity condition of the coefficient uk in A, implied by the condition (2.1). This condition provides the weights (deg α = 1, deg β = 2), implying for deg uk = k − 1 the homogeneity relation s + 2t = k − 1 for the degrees s and t of the monomials of the formula (2.3) for the quantity uk . For instance, for k = 6, the Pascal triangle provides the following coefficients of formula (2.3):
Chapter 2
17
1 1 1
2
1 1 1
1 1
3 4
3 6
5
10
u6 1 4
10
1 5
1
Therefore, the quantity u6 is represented as the sum of three monomials of weight 5: u6 = 3αβ 2 + 4α3 β + 1α5 . Using this algorithm, I calculated the places of the 24 nonzero elements A of the field, consisting of 25 elements in half an hour. The resulting table of this field is k
p=5
u 4 3 2 1 0
13 7 19 1 ∞ 0
15 10 11 8 24 1
5 9 2 4 18 2
16 14 21 17 6 3
20 23 22 3 12 4 v
Example. A10 = 3 · A + 1 · 1, A19 + A8 = A10 . Remark. The symmetry center, denoted by the sign “◦” has the following property (easy to prove): k − ` = 12 (mod 24) whenever the symbols’ k and ` places are situated symmetrically with respect to this center (on the finite torus). For instance, 21 − 9 = 12, 17 − 5 = 12, 24 − 12 = 12 (it would be equal to (z − 1)/2 for the field of z elements). The reason of this symmetry is the evident identity A12 = −1 (that is, u12 = 0, v12 = 4). The symmetry allows one to reduce the field’s table calculation, making it two times faster: it suffices to find the coordinates (uk , vk ) of the symbols k ≤ z/2 for the case of the field having z = pn elements.
18
Galois field’s table.
The field’s table may be interpreted the following way. The multiplication of the elements of the field by A acts on Ak as a linear operator on the plane of the table: Ak · 1 = u k A + v k 1 , Ak · A = uk+1 A + vk+1 1 .
Therefore, the matrix of this linear operator on the plane with coordinates u and v, equipped with the basis (1, A), has the form vk vk+1 k (A ) = . uk uk+1 For k = 1, this matrix is equal to 0 2 0 β for p = 5 . equal to (A) = 1 2 1 α The relation (2.1) is simply the characteristic equation for matrix (A). The operator of the multiplication by Ak being the k-th power of the multiplication by A, the matrix (Ak ) is the k-th power of the matrix (A). Therefore, the field’s table construction provides a representation of the field, consisting of p2 elements, by the second order matrices (Ak ), whose elements belong to Zp (being residues for the division by p). The field’s operations are represented as the matrices addition and multiplication (A)k · (A)` = (A)k+` , (A)k + (A)` = (A)k∗` .
For the field, consisting of z = pn elements, a similar construction provides a field representation by the matrices of order n with elements in the field Zp . Some examples, where n = 3, are listed below, in §8. For the fields of p2 elements, where p = 7, 11 end 13 the calculations, quite similar to those described above for p = 5, provide the following answers: p=7 (A) =
0 2 1 2
p = 11
(A) =
0 3 1 1
p = 13
(A) =
0 2 1 4
.
The resulting table of the field of p2 = 49 elements fills the finite torus Z27 by the following residues (modulo 48):
Chapter 2
19 u 6 5 4 3 2 1 0
p=7
25 9 17 41 33 1 ∞ 0
44 35 37 23 38 18 48 1
7 30 28 29 22 36 19 10 2 15 21 27 32 40 2 3
3 39 34 12 5 6 16 4
45 42 26 14 43 47 46 13 4 11 31 20 8 24 5 6 v
The table of the field of p2 = 121 elements fills the finite torus Z211 by the following residues (modulo 120):
u 10 61 11 15 76 22 43 78 53 62 56 105 9
25 42 95 17 99 26 40 20 106 69
8 13 94 14 83 115 8 7 109 4
p = 11
87 30 57 28
7 5
6 104 59 70 101 33 63 91 110
6 37 32 29 19 52 107 81 38 54 118 111 5 97 51 58 114 98 21 47 112 79 89 92 4 49 50 31 3
93 41 10 119 44 66 64
3 73 65 88 117 90 27 68 55 23 74 34 2
35 67 9
1
1
0
∞ 120 84 72 48 96 36 108 12 24 60 0
46 80 100 86 39 77 35 102
45 116 2 113 18 103 82 16 75 71 1
2
3
4
5
6
7
8
9
10
v
Bold numbers in these tables represent the multiplicative generators Ak of the multiplicative groups of the fields (corresponding to these values of k, which are relatively prime to z − 1 = p2 − 1 = 120 in our case p = 11). The table of the field of p2 = 169 elements fills the finite torus Z213 by the following residues (modulo 168 = 23 · 3 · 7):
20
Galois field’s table.
u 12 35 45 161 165 13 76 58 47 158 122 23 166 64 11 15 145 143 88 91 52 95 121 111 96
6 162 156
10 141 10 132 101 79 114 49 54 103 53 120 46 69 9 113 51 75 41 73 26 18 104 21 92 150 86 25 8 127 32 39 40 100 87 164 55 106 89 35 65 118
p = 13
7 71 152 108 33 62 151 31 50
9 144 44 167 147
6 155 63 83 128 60 93 134 115 67 146 117 24 68 5 43 34 149 119 5 4 29 109 2
66
22 139 80
3
16 124 123 116
8 105 20 102 110 157 125 159 135
3
57 153 130 36 137 19 138 133 30 163 17 48 94
2
99 72 78 90 12 27 37 11 136 7
1
1 148 82 107 38 74 131 142 160 97 81 77 129
0
∞ 168 98 56 28 42 154 70 126 112 140 14 84 0
1
2
3
4
5
6
7
8
9
4 59 61
10 11 12
v
Remark. While the field is defined unambiguously by its elements number, the table of this field is not defined by it, being dependent of the choice of the multiplicative generator A of the group of the nonzero elements of the field. Instead of the generator A, one might choose a different primitive element, A˜ = Ak (which is primitive just when k is relatively prime to the number z − 1 for a field of z elements). The primitive elements form the “Euler group” Γ(z − 1). We discuss the influence of the choice of the primitive element A on the constructions described below in §7: these investigations lead to some astonishing facts of the projective geometry of finite sets (see §6 and §7 below).
Chapter 3 Chaoticity and randomness of Galois field table’s numbers Looking at the preceding tables of Galois fields, one has the impression, that their fillings by the integers from 1 to z − 1 (z being the fields’ elements number, equal to p2 in our examples) behave like some kind of random numbers tables: it is difficult to guess the place of the next symbol k + 1, knowing the preceding symbol’s k place. The attempts to formulate this natural science observation as a mathematical statement lead to hundreds of conjectures. Perhaps, most of these conjectures will become interesting proved theorems in the future (while at present it had happened only to few of these conjectures). The general scheme of the “randomness” conjectures formulations is the following reasoning. The genuine random fillings have several properties, studied in probability theory and in the stochastic processes theory. To check the “randomness” of the filling of the table of the field, choose one of these properties, and verify, whether the quasirandom numbers, filling the table, approximatively do verify it. The conjecture, to which one arrives this way, claims, that the randomness criterium, that one had chosen, is approximatively fulfilled by the matrix of the field of z = pn elements better and better, while the prime number p (for a fixed n) or the elements number z is growing. The limiting relation (for p → ∞) is conjectured to be fulfilled exactly. Thus, to fix the mathematical formulation of the pseudo-randomness conjecture one has to describe exactly the chosen criterium. Since there exist 21
22
Chaoticity of field’s table.
many such criteria, one gets many conjectures. I shall discuss below a short list of simplest examples (which are already nontrivial and interesting, in spite of their simplicity: every highschool student might check these conjectures empirically, fixing p, even using no computer’s help). Let us start from an example: suppose, that the whole table is subdivided into two disjoint parts, and count those numbers k, filling the table, which occur in the first part, G. For a random filling the number N of the occurrences of G among m trials is proportional to the volume (to the area in the 2-dimensional case) of the domain G: N |G| ≈ , m z (where z is the total volume of the table, that is z = p2 in the case of the field of p2 elements). Of course, for m = z the approximated equality, written above, is exactly fulfilled by the sequence of the field’s elements Ak , 1 ≤ k ≤ m, since each cell of the table, belonging to the domain G (supposed to consist of table’s cells) is visited by the sequence {Ak } just once. Therefore, to formulate a nontrivial conjecture on the equidistribution of the elements {Ak } along the field one should take only a part of the whole sequence. Choose for this a number ϑ strictly between 0 and 1 and consider the beginning of the sequence {Ak }, consisting of m ≈ ϑz members (1 ≤ k ≤ m). Then we get the following Geometrical progression equidistribution conjecture for the Galois field of z = pn elements: |G| N = , (3.1) lim p→∞ m z where N is the number of those first m elements of the sequence {Ak } of elements of the field of z elements, which do belong to the domain G. I had fixed here the table’s dimension n, but one might consider also the z → ∞ limits, where n is not fixed. Example. For the field of z = p2 = 25 elements, choose as G the union of the first two columns of the table: |G| = 10, v = 0 or 1 in G. Let us take the first half of the geometrical progression {Ak }, 1 ≤ k ≤ (12 = m). Then the visits number (counted by the above table of the field, page 17) is N = 5. The deviation from the theoretical randomness criterium (3.1) is
Chapter 3
23
in this case
N |G| 5 10 1 − = − = . m z 12 25 60 Therefore, even while the prime number p = 5 is not so large, the approximation to the equidistribution is rather good. I had not proved the limit theorem (3.1) in its general form, but I discuss below (in §4) some of its versions, which are proved. As other randomness criteria one might suggest, for instance, the following ones. Subdivide the field in two disjoint domains F =G∪H , whose elements numbers are, correspondingly, |F | = z ,
|G| = rz ,
|H| = sz ,
where r + s = 1. For a genuinely random sequence of the choices of the field’s elements Ak in F the frequencies of the jumps from G to G, from G to H, from H to G and from H to H for the transition to the next element of the sequence are, correspondingly, r 2 , rs, sr and s2 . For the geometrical progression {Ak } (which one might leave unshortened in this case: 1 ≤ k < z) one expects similar frequencies of the four events (Ak ∈ G, Ak+` ∈ G), (Ak ∈ G, Ak+` ∈ H) and so on. Geometrical progression {Ak } mixing conjecture for the z = pn elements field. The numbers N (G, G), N (G, H), N (H, G) and N (H, H) of the occurrences of the k, for which (Ak ∈ G, Ak+` ∈ G), (Ak ∈ G, Ak+` ∈ H), and so on are asymptotically proportional to the frequencies (r 2 , rs, sr, s2 ) of the random jumps: lim
p→∞ `→∞
N (G, G) = r2 , z
lim
p→∞ `→∞
N (G, H) = rs , z
...
.
One more randomness criterium is provided by the table variation, which measures the differences of the symbols k at the neighbouring cells of the
24
Chaoticity of field’s table.
table. For instance, one might counts the sum Σ of the differences |k − `|, or better the sum of the distances %(k, `) of the neighbouring symbols in the table, comparing it with the similar sum for a purely random filling of the table by the symbols k from 1 to z. The asymptotical difference of every of these 2 sums for the field (which differ) from their mathematical expectation for a random filling is expected to reach (relatively) small values for large primes p (or for the growing numbers z = pn of elements of the fields). For the table of the field having p2 = 25 elements (at page 21), the observed averaged distance %(k, l) between the residues k, l occupying the neighbouring places in the toric table, is 6.41. The random filling expectation of this distance (between the residues modulo p2 − 1) is (p2 − 1)/4 = 6. For the table of the field, having p2 = 169 elements (at page 25) the observed averaged distance %(k, l) between the residues k, l, horizontal neighbouring in the table, equals 42.0299, the random filling expectation being (p2 − 1)/4 = 42. The neighbouring in the table is to be considered here accordingly to the torical geometry (say, for p = 5 the value u = 4 is a neighboor of u = 0(≡ 5)). In a similar way, one might consider a different kind variation, measuring the distances of the places of Pthe symbols k and k +1 in the table. One might consider the quantity % = k %(k, k + 1), summing either for the cyclically closed sequence, or for the usual sequence, 1 ≤ k < z. The averaged distance %(k, k + 1) between the places of the neighbouring residues in the toric p × p table of the field, having p2 = 25 elements, is 13 , the random filling providing the expected average p/2 = observed to be 2 24 2 21 . The distance between the places was measured using the sum of the difference of the coordinates (using the torus geometry, that is, considering the coordinates as being residues modulo p). Counting the summands of this sum one should not forget the torical geometry of the table, filling Znp by the symbols k. For instance, for p = 7 and n = 1 consider the filling of the 7 consecutive places of the table by the cyclical sequence of values k = (1, 5, 4, 2, 3, 7, 6) (figure 3.1). For this filling one gets % = 3 + 1 + 2 + 1 + 2 + 1 + 2 = 11 , Σ = 4 + 1 + 2 + 1 + 3 + 1 + 2 = 14 , since %(3, 7) = 3 and %(6, 1) = 2 in the torical geometry of Z7 .
Chapter 3
25 2 3 1 4 7 5
6
Figure 3.1: Distance 2 between points of a finite torus. The variation % takes the value 11 on this cyclical sequence of the seven residues modulo 7. For these variations the field’s table randomness conjecture suggests the relatively small difference between the quantity %, calculated from the table of z = pn elements field, and the mathematical expectation of a similar sum for a genuinely random filling of the table, provided that p (or z) is sufficiently large. One might use here either the sum of the m distances between the m elements of a cyclical sequence, or the sum of the m − 1 distances between the elements of an ordinary sequence of m elements. As one more randomness characteristics of the set {Ak } in the table one might use such quantity, as the minimal radius r(m) of the balls, centered at the first m points of the set, which cover together all the table, or one might consider the maximal radius R(m) of the ball, containing no points of this subset (figure 3.2).
R r
Figure 3.2: Covering balls and void balls of a set of points One should compare the values r(m) and R(m), calculated from the field’s tables, with the similar characteristics of m genuinely random points: the field table chaoticity conjecture claims the similarity of these quantities behaviours
26
Chaoticity of field’s table.
for the fields of z = pn elements (where p → ∞ or z → ∞), provided that m ≈ ϑz (where 0 < ϑ < 1 is fixed). One more randomness characteristic of a set of m points of the table is the percolation radius, defined the following way. Enclose each point of the set in a radius r ball, centered at this point. If r is sufficiently small, one can’t cross the table from one side to the opposite one (creating an uncontractable path on the torus) along these small balls union. If the radius is sufficiently large, such a “percolation” through the union of the balls, representing the defects of the material, becomes possible (the word “percolation” is borrowed from the leaks studies in the materials of the vessels) – see figure 3.3.
no percolation
percolation
Figure 3.3: Percolation appearence, due to the defects’ radius growth. The critical (minimal) value r(m), first providing the percolation apparence, is called the percolation radius of a given set of m points: it is the smallest radius of the defects, producing the leakage. The percolation chaoticity conjecture for the points of the geometrical progression {Ak } in the field’s table compares the percolation radius behaviour r(m) for m independent random points of the table (say, for m ≈ ϑz, where 0 < ϑ < 1 is fixed and the field contains a large number z = pn of elements). Here, as above, I mean the limiting behaviour for p → ∞ (but the z → ∞ limit might also be considered). In these percolation studies it might be also interesting to replace the balls of radius r by the segments of the progression {Ak : |k − k0 | ≤ %} , comparing it with a similar set of segments of a random sequence {Ak } of m points of the field’s table. Defining this way the quasi-percolation radii %(m), their conjectured behaviours should be similar for the first m ≈ ϑz points
Chapter 3
27
{Ak } of the z = pn elements field’s table and for the random sequence of m points of the table (where, as usually, 0 < ϑ < 1 is fixed and p → ∞ (either z → ∞)). Of course, one is able to invent many different criteria of the table chaoticity, and every one of them leads to an (interesting?) conjecture of ergodic character, which deserves to be studied empirically (and which would, hopefully, become a theorem, if the numerical experiences would confirm it, which theorem one might later try to prove). The resulting theory is some number-theoretical finitistic version of the ergodic theory of the tori automorphisms, where the chaoticity and the mixing properties of the progressions {Ak } had been studied for the volumes preserving automorphisms A of the continuous torus T n . The difference of our case depends on the fact, that the finite torus Znp consists of a finite number z of points, and that the infinite time limit, used in the ergodic theory to define the time average, is replaced in our case by the limit for the growing number m ≈ ϑz of the points in the orbit of the dynamical system which we are studying (occurring due to the growing of the parameter p or of the number z = pn of the points of the finite torus). It is interesting, that the percolation chaoticity problem had not been studied, as far as I know, even in the (more simple) case of the continuous torus hyperbolic automorphisms of ergodic theory.
28
Chaoticity of field’s table.
Chapter 4 Equipartition of geometrical progressions along a finite one-dimensional torus There exists two very different ways to formulate a problem: the French way consists in the most general formulation, leaving no possibilities for further generalizations (differing from the nonsense), the opposite Russian way is to choose the simplest case which can not be simplified further (preserving some content of the problem)1 . I had tried above to formulate the field’s tables randomness conjectures in the French form. Let us consider now the Russian form of the first of these conjectures, claiming the equidistribution of the progression in the field of pn elements. To do it, we restrict ourselves, considering the simplest field Zp (consisting of the p residues of the division by a prime number p), that is consider the simplest case n = 1 of the general theory of §3. To simplify the formulae, we shall suppose the prime number p to be odd, and as the domain G of §3 we shall chose the first half of the nonzero elements of the field, {c : 1 ≤ c ≤ (p − 1)/2}, |G| = (p − 1)/2. As the segment of the geometrical progression of residues we consider its first m = (p − 3)/2 terms, {Ak : 1 ≤ k ≤ m}. 1
Tchebyshev, who had a lot of friendly relations to French mathematicians, like Liouville, had never discussed with them any mathematics, to make no harm to his Russian approach by their influence, as he described it, returning home.
29
30
Geometric progressions equipartition.
This strange choice of the “half” of the progression (ϑ ≈ 1/2) is explained by the little Fermat theorem statement, Ap−1 = 1, making Ak = −1 for k = (p − 1)/2. Therefore this term of the progression is not random at all, while the randomness might be expected for smaller k, that is for k ≤ (p − 3)/2. Calculating these segments of the progressions of residues modulo p for p = 5, 7, 11 and 13, we should first find for every p all the primitive elements A (for which the smallest period T of the progression takes just the Fermat theorem value T = p − 1, while for other A it might be a smaller divisor of p − 1). These progressions and periods for p = 5 are provided by the following table: A 1 2 3 4
{Ak } T 1, 1, 1, 1 1 2, 4, 3, 1 4 3, 4, 2, 1 4 4, 1, 4, 1 2
N
Σ
1 0
2+1+2+1=6 1+2+1+2=6
The primitive elements are here A = 2 and A = 3; they are denoted by the bold characters. The column N represents the number of the visits of the segment of the first m = (p − 3)/2 terms of the progression to the domain G (consisting of the residues, which do not exceed (p − 1)/2). For p = 5 we obtain m = 1, (p − 1)/2 = 2, and therefore N (A = 2) = 1 and N (A = 3) = 0. The column Σ represents the sum of the distances in Z5 between the consecutive members of the progression (considering the progression as a cyclic sequence, that is including also the distance from the last member of the period to the first one). The differences between the observed frequencies and the space average (measuring the error of the equipartition conjecture for the geometric progression’s residues in the space of the nonzero residues) are equal to: A=2 :
|G| 1 2 1 N − = − = , m z−1 1 4 2
N |G| 0 2 1 − = − =− , m z−1 1 4 2 We observe that in both cases the error of the approximation to the equidistribution has the absolute value 1/2. In the average (with respect to A=3 :
Chapter 4
31
the choice of the primitive element A) the equipartition criterium is fulfilled exactly |G| N = , m z−1 where N= =
X
N (A) /(the number of the primitive elements A) =
(1 + 0) . 2
Similar calculations for the prime number p = 7 provide the following table of the answers (p = 7, m = 2, |G| = 3, z = 7): A 1 2 3 4 5 6
{Ak } 1, 1, . . . 2, 4, 1, . . . 3, 2, 6, 4, 5, 1, . . . 4, 2, 1, . . . 5, 4, 6, 2, 3, 1, . . . 6, 1, 6, 1, . . .
T 1 3 6 3 6 2
N
Σ
2
1 + 3 + 2 + 1 + 3 + 2 = 12
0
1 + 2 + 3 + 1 + 2 + 3 = 12
(we used the fact, that the distance between the elements 1 and 5 of Z7 equals 3). Thus, the mean number of visits of G is equal to N = (2 + 0)/2 = 1, therefore, N /m = 1/2. The spatial average also equals to |G| 3 1 = = . z−1 6 2 Therefore, the equidistribution criterion is once more fulfilled exactly (in the average with respect to the primitive element A choice), like for p = 5. The answers for the case p = 11 take the form p = z = 11, m = 4, |G| = 5, z − 1 = 10,
32
Geometric progressions equipartition. A 1 2 3 4 5 6 7 8 9 10
{Ak } 1, 1, . . . 2, 4, 8, 5, 10, 9, 7, 3, 6, 1 3, 9, 5, 4, 1, . . . 4, 5, 9, 3, 1, . . . 5, 3, 4, 9, 1, . . . 6, 3, 7, 9, 10, 5, 8, 4, 2, 1 7, 5, 2, 3, 10, 4, 6, 9, 8, 1 8, 9, 6, 4, 10, 3, 2, 5, 7, 1 9, 4, 3, 5, 1, . . . 10, 1, 10, 1, . . .
T 1 10 5 5 5 10 10 10 5 2
N
Σ
3
30
1 3 1
30 30 30
We see from this table, that the visits statistics provides the values
N=
3+1+3+1 =2, 4
1 N = , m 2
the space average being equal to
|G| 5 1 = = . z−1 10 2 Thus the equipartition criterium is fulfilled exactly in the average under the choice of the primitive element A (the absolute values of the error for the particular choices of A being all equal to 1/4).
The answers for the case p = 13 provide the values z = 13, m = 5, |G| = 6, z − 1 = 12. The progression table takes in the case p = 13 the form
Chapter 4
33 A 1 2 3 4 5 6 7 8 9 10 11 12
{Ak } 1, 1, . . . 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, . . . 3, 9, 1, . . . 4, 3, 12, 9, 3, 1, . . . 5, 12, 8, 1, . . . 6, 10, 8, 9, 2, 12, 7, 3, 5, 4, 11, 1, . . . 7, 10, 5, 9, 11, 12, 6, 3, 8, 4, 2, 1, . . . 8, 12, 5, 1, . . . 9, 3, 1, . . . 10, 9, 12, 3, 4, 1, . . . 11, 4, 5, 3, 7, 12, 2, 9, 8, 10, 6, 1, . . . 12, 1, 12, 1, . . .
T 4 12 3 6 4 12 12 4 3 6 12 2
N
Σ
4
42
2 1
42 42
3
42
In this case the visits number, averaged along the 4 primitive elements of the field, equals 4+2+1+3 1 N= =2 , 4 2 therefore N /m = 1/2. The space average also takes the value |G| 6 1 = = . z−1 12 2 Thus, for p = 13, the equipartition criterium is also fulfilled exactly (in the average with respect to the choice of the primitive element A). The individual choices (A = 2, 6, 7, 11) provide, correspondingly, the errors (3/10, −1/10, −3/10, 1/10) . These empirical studies lead us to the following conclusion. Theorem. The equipartition criterium for the distribution of the first m = (p − 3)/2 members of the progression {Ak } distribution among the nonzero residues of the division by p is exactly fulfilled (in the average with respect to the choice of the primitive element A) for the domain G = {1 ≤ c ≤ (p − 1)/2 = |G|} in the field Zp (for any odd prime number p).
34
Geometric progressions equipartition. In other words, for the averaged visits number P N (A) , N= (number of the primitive elements A)
we have the “ergodic” value N |G| 1 = = . m p−1 2 Proof. Together with a primitive element A the inverse residue modulo p, B = A−1 , is also a primitive element. Lemma. The following identity holds: N (A) + N (B) = m . Proof of the Lemma. Taking into account the Fermat congruence Ap−1 = A2m+2 = 1, we deduce that the two sequences {1 ≤ k ≤ m} and {1 ≤ ` ≤ m} of the progressions Ak and B ` = Ap−1−` cover with multiplicity one all the progression {Ai , 1 ≤ i ≤ p − 1}, except its two trivial (“nonrandom”) terms, Ap−1 = 1 and Am+1 = −1. Therefore they cover (with multiplicity 1) every element c of domain G, {2, 3, . . . , m + 1}, except the element c = 1. Thus the sum N (A) + N (B) of the visits numbers for both sequences to domain G equals m, and the Lemma is therefore proved. The Lemma implies the equality of the mean (over the choices of primitive element A) visits number N to m = (p − 3)/2. Indeed, the whole set of the primitive residues A consists of α (disjoint) pairs of the form {A, B}, where AB = 1 P (as a residue modulo p). Each pair provides the contribution m to the sum N (A), accordingly to the Lemma. Therefore, the whole sum equals αm, whence we find the averaged number of visits, P N (A) mα m N= = = . 2α 2α 2 Thus, N /m = 1/2, which proves the Theorem’s statement.
Chapter 4
35
tables show, that in all our examples the variation Σ = P Thek above k+1 %(A , A ) of the whole progression of p−1 residues, considered as points on the finite circle Zp , equals the value Σ=
p2 − 1 , 4
independently of the choice of the primitive residue A. The mean variation Σ of a random cyclical sequence of p − 1 = 4 points of Z5 can easily be calculated. It suffices to consider the 6 sequences, starting with the residue modulo 1, (1, 2, 3, 4) , (1, 2, 4, 3) , (1, 3, 2, 4) , (1, 3, 4, 2) , (1, 4, 2, 3) , (1, 4, 3, 2) . Their variations Σ are, correspondingly, equal to (5, 6, 7, 6, 7, 5) (we had used the distance %(4, 1) = 2 in Z5 ). Therefore, the mean value Σ of the variation of a cyclical sequence of 4 points on the finite circle Z5 is Σ = 6. Thus, the variations of the cyclical geometrical progressions, formed by the powers of the primitive residues A (of the division by p = 5), calculated above, Σ(A) = 6, coincide with the mean variation Σ = 6 of a random cyclical sequence (of the same length p − 1 = 4) of elements of Z5 . This observation provides one more argument for the quasirandomness statement (of the table of the field of p = 5 elements). When the prime number p is growing, the mean variation Σ of a random sequence of p − 1 points on the finite circle Zp grows like p2 /4. This follows from the argument below. The distance between two randomly choosen points of this finite circle attains the values from 1 to (p−1)/2. Its mean value is easily calculated to be close to p/4. Therefore, the sum of all such distances between the consecutive points of our sequence (there are p − 1 such distances) is asymptotically growing with p like p2 /4 (with a declining relative error). Thus, the calculations of the variations Σ of the cyclical geometrical progressions of lengths p − 1 in the fields Zp (for p ≤ 13), presented above, confirm once more the quasirandomness of the table of the finite field with p elements. Remark (on logarithm’s complexity). The chaoticity of the distribution of the geometric progression of the residues leads to interesting facts and conjectures of complexity theory. If a is a primitive residue modulo p, each
36
Geometric progressions equipartition.
nonzero residue x modulo p has the form ak , and the complexity conjecture is, that to calculate the “logarithm” k of x is a difficult computational problem. To define the difficulty measure, one might classify functions ( on a finite set with a finite number of values) according to the “degree of complexity” of the formula, defining this function. To define numbers, measuring complexity, I shall consider the simplest case of binary functions f : (Z/nZ) → (Z/2Z). Such a function can be considered as a sequence (x1 , . . . , xn ) of n elements, each of them being either 0 or 1. There are 2n such sequences, and they form the modulo 2 vector-space (Z2 )n . One can consider these functions as the vertices of a cube of dimension n. To measure the complexity of a function x following Newton’s idea, we associate to it the first difference function, y, defined as the sequence of the binary residues y(k) = x(k + 1) − x(k) (mod 2) . (The argument k being a residue modulo n, we get n differences of n residues, considering the next element after the last one to be the starting element of the sequence. Making the sequence cyclic, we avoid the boundary effects). Thus, if x = (1, 0, 0, 1, 1) we obtain the differences y = (1, 0, 1, 0, 0) (since (y5 = x1 − x5 = 0). The differences operator is a linear operator (abelian group homomorphism) A : Zn2 −→ Zn2 .
The complexity of a point x ∈ Zn2 will be calculated in terms of the sequence of the consecutive differences, At (x) ∈ Zn2
(t = 1, 2, . . . ) .
Example. For the constant function x we get A(x) = At (x) = 0. For a polynomial x of degree d (that is, for x(k) = a0 k d + · · · + ad ) we get At x = 0 for any t > d. We shall study below the spectral properties of the linear operator A. It is natural to consider the constants to be the simplest functions, the small degree polynomials to be less complicated, than the higher degree polynomials, and to consider the nonpolynomial functions as even more complicated objects. (I shall not formulate the evident next steps, involving the
Chapter 4
37
exponents and the differential equations solutions – the reader might construct his hierarchy of more and more complicate functions x according to his needs). The conjecture is, that the the logarithmic functions, defined above, are complicate. I shall not prove this conjecture, but I shall show several examples, confirming it. Other conjectures claim, that most of the 2n functions, forming Zn2 , are like random sequences (at least asymptotically for n → ∞ and at least for the majority of these functions). I shall not prove it, but the examples, discussed below, provide the complicate functions, behaving similarly to the random sequences in the numerical experiments. I hope, that this quasirandom behaviour is rather a general phenomenon, than a special property of our examples. To understand the complexity ranges, we start from the general study of the differences operation A : Zn2 → Zn2 . Being a mapping of a finite set to itself, the operator A decomposes the set Zn2 into connected invariant components. We consider these components as directed graphs (which edge leaving x connect point x with Ax). Each connected component of the graph of a mapping consists of an attracting cycle Om of some length m ≥ 1:
O1 =
, O2 =
, O3 =
, ... ,
and of the trees, attracted by the vertices of the cycle. We shall need the binary trees T2q of 2q vertices:
T2 =
, T4 =
, T8 =
, ... .
We shall denote by Om ∗ T2q the component, whose cycle Om is attracting a tree T2q at each vertex: O1 ∗ T2q = T2q , and next “products” are
O3 ∗ T2 =
,
O2 ∗ T 4 =
, ... .
38
Geometric progressions equipartition.
Theorem 1. The graph of the differences operator A : Zn2 → Zn2 has the form, presented in the following table, for n ≤ 12: n
number of the components
cycles and trees
A u = Av
2 3 4 5 6 7 8 9 10 11 12
1 2 1 2 4 10 1 6 10 4 24
(O1 ∗ T4 ) (O3 ∗ T2 ) + (O1 ∗ T2 ) (O1 ∗ T16 ) (O15 ∗ T2 ) + (O1 ∗ T2 ) 2(O6 ∗ T4 ) + (O3 ∗ T4 ) + (O1 ∗ T4 ) 9(O7 ∗ T4 ) + (O1 ∗ T2 ) (O1 ∗ T256 ) 4(O63 ∗ T2 ) + (O3 ∗ T2 ) + (O1 ∗ T2 ) 8(O30 ∗ T4 ) + (O15 ∗ T4 ) + (O1 ∗ T4 ) 3(O341 ∗ T2 ) + (O1 ∗ T2 ) 20(O12 ∗ T16 ) + 2(O6 ∗ T16 ) + (O3 ∗ T16 ) + (O1 ∗ T16 )
A2 = 0 A4 = A A4 = 0 A16 = A A8 = A 2 A8 = A A8 = 0 A64 = A A32 = A2 A342 = A A16 = A4
The proof is a direct verification. Say, for n = 2 there are 2n = 4 vertices, and the differences operator is, by definition, acting the following way: A(0, 0) = (0, 0) , A(0, 1) = (1, 1) , A(1, 0) = (1, 1) , A(1, 1) = (0, 0) , providing the graph, whose only component is
(0, 1) T4 :
(1, 1)
(0, 0)
(1, 0)
(4.1)
Denoting by δ the shift operator (δx)k := xk+1 , we obtain the formulas A = 1 + δ, δ n = 1. Therefore, we get, for n = 3, A = 1 + δ , A2 = 1 + 2δ + δ 2 = 1 + δ 2 , A3 = 1 + δ + δ 2 + δ 3 = δ + δ 2 A4 = δ + δ 2 + δ 2 + δ 3 = δ + 1 = A . Several evident properties of the formulas in the table are easily provable in the general situation. Thus, for n = 2m one has An = 0, since all the binomial coefficients Cni are even for i 6= 0, n: An = 1 + δ n = 1 + 1 = 0 (mod 2) .
Chapter 4
39
Now we shall study the places of the logarithmic functions in this table. The explicit calculations show, that their complexities are close to the maximal possible complexity of a binary function (for a given value of n). For a primitive residue a modulo p we define the “logarithm of the residue k” by the Fermat formula aloga (k) = k (mod p) . Reducing this integer modulo 2, we construct the binary function with the values x(k) = loga (k) (mod 2) ∈ Z/(2Z)
(for the argument’s values k = 1, 2, . . . , p − 1).
Example. p = 7, a = 3 provide log3 k = (0, 2, 1, 4, 5, 3), x(k) = (0, 0, 1, 0, 1, 1). Thus we obtain for each a the sequence of the n = p − 1 logarithms reduced modulo 2, x ∈ Zn2 . Now we shall apply the complexity, defined by the graphs of Theorem 1, to this binary sequence x. Theorem 2. The modulo 2 reduced logarithms x of the consecutive residues modulo p have the following values (for p ≤ 13) in terms of the differences graphs of the preceding theorem (for n = p − 1) x=1
(x ∈ T4 , a = 2);
p=3 x=6
(x ∈ T8 , a = 2 or 3);
p=5
x = 11
p=7
p = 11
216
285 360
O30
(x ∈ T16 , a = 3 or 5);
(x = 285 ∈ (O30 ∗ T16), a = 2, 6, 7 or 8);
40
Geometric progressions equipartition.
3546
p = 13
1206
1238 2857
1647 2488
1935 3450
4050 2737
O12
O12
(x = 1238 ∈ (O12 ∗ T16), a = 2);
(x = 1206 ∈ (O12 ∗ T16), a = 11).
In this description we denote the binary sequence x = (x1 , . . . , xn ) ∈ Zn2 by the “binary decimal” integer 2n−1 x1 + 2n−2 x2 + · · · + xn (thus, x = 285 in the case p = 11, n = 10 means the sequence (0, 1, 0, 0, 0, 1, 1, 1, 0, 1) of the 10 binary digits of the number 285 = 256 + 16 + 8 + 4 + 1). The proofs of the statements of Theorem 2 are finite, but long calculations. In the simplest case p = 3, a = 2 the geometrical progression {2k } = 1, 2 (mod 3) for k = 1, 2 implies the logarithms log2 1 = 0 ,
log2 2 = 1 .
The reduced logarithms sequence is (x1 = 0, x2 = 1), providing the position of x in the graph of the theorem, according to formula (4.1). For higher values of p (especially for p = 11 and 13) the calculations are longer. To accelerate them it is useful to reduce the length n cyclical sequences x modulo the n cyclic rotations group Z/(nZ) (identifying, say, the sequences (0100) and (0001) for n = 4). For this reduction by the action of δ it is useful to consider x as a residue modulo 2n − 1. Since the operator A commutes with these rotations, this reduction accelerates the calculations (about n times). It is also useful to calculate first the kernel of At (say, for large t). This kernel is represented by the vertices, forming a binary tree. To obtain the attracted trees of the cycles it is sufficient to add to the points of the cycle this subspace of Zn2 . This reasoning explains the homogeneity of the graphs of the table of Theorem 1: all the attracted trees are
Chapter 4
41
isomorphic to the above kernel (the union of the cycles being the image of the linear operator At for large t). These long calculations lead to the table of Theorem 2, whose comparison with the table of Theorem 1 shows, that the complexity of the logarithmic function almost attains the maximal value, possible for any binary function on a set of n points. This observation follows from both theorems only in the case n ≤ 12, but it might be considered for larger values of n, at least as a probable conjecture. Similar theorems (and conjectures) should be also considered for the nonbinary functions, for instance, for the functions, whose values are residues modulo some integer q: x ∈ (Z/qZ)n ,
xk ∈ Z q .
Unfortunately, I can’t guess the answers in such a generalization of Theorem 1 (even for q = 2, as above): the table of Theorem 1 shows a rather irregular dependence on n, and even the averaged asymptotics for n → ∞ are interesting, but unknown. All this set of theorems and conjectures can be extended to the general Galois fields almost literally, but I had restricted myself above to the simplest case of the residues field Zp (and sometimes only to the binary base Z2 ), since the missing complexity theory should first study this elementary case. The author is grateful to Prof. Shparlinski (Departement of Computing, Macquarie University, Sydney), who, having read the original draft of the present book, had proved some of the conjectures, discussed above, correcting also some misprints in my previous publications of some others.
42
Geometric progressions equipartition.
Chapter 5 Adiabatic study of distribution of geometrical progressions of residues I shall describe here some physical arguments, explaining the asymptotical equipartition of the sequence of the residues of the division by p of the members of the geometrical progression {Ak , 1 ≤ k ≤ ϑp} among all the nonzero residues of the division by p, A being a primitive residue and the number ϑ, 0 < ϑ < 1, being fixed, while p tends to infinity. We shall try to evaluate the number N of those residues of the progression’s members, which belong to the interval G :
1 ≤ c ≤ µp
of the values of the residues c. To make it, we shall use the logarithms, transforming the geometrical progression into an arithmetical one: ln Ak = `k = ka
(where a = ln A) .
The condition Ak (mod p) ∈ G can be written in terms of the logarithms as the inclusion of the number `k into one of the intervals of the following system (represented in figure 5.1): [ `k ∈ ∆j , 43
44
Adiabatic study of equipartition.
where ∆j is the interval between the logarithms of members of two arithmetical progressions, ∆j = {` : ln(jp) < ` ≤ ln(jp + µp)} . The length of the interval ∆j equals jp + µp µ µ j+µ |∆j | = ln = ln 1 + ≈ = ln jp j j j (for large j). `k,min
ka
∆
j ln(pj)
`k,max
(k + 1)a
∆j+1
ln(p(j + 1)) ln(p(j + µ))
Figure 5.1: Adiabatic approximation of a nonuniform sequence of the logarithms of the members of an arithmetical progression. PThe sum D(µ) of the lengths of all these intervals is a quantity of order µ (1/j), where the numbers j of the intervals, forming this finite sum, is fixed by the maximal and minimal logarithms `k of the members of our geometrical progression. This leads to an approximate relation D(µ) ∼ µ ln(jmax )
(where jmax → ∞ for p → ∞) ,
the “approximation” word meaning the smallness of the relative error. The total length of the whole interval of the axis `, containing all our logarithms `k , is D(µ = 1) ∼ ln(jmax ). The arithmetical progression {`k } is uniformly distributed along the ` axis (according to the H. Weyl theorem on the equipartition of the fractional parts of an arithmetical progression in the interval (0, 1)). This leads to the guess, that the number N of the visits of the points `k to the union of the intervals ∆j should be asymptotically proportional to the
Chapter 5
45
fraction, formed by the lengths sum of these intervals in the length of the whole segment of the axis `, that is it should be asymptotically proportional to the ratio D(µ)/D(1). We arrive, therefore, to the conclusion, that for p → ∞ one should expect the asymptotically behaviour of the visits number D(µ) N −→ ∼µ , ϑp D(1) which means the asymptotical equipartition of the sequence of the residues of the division of the members {Ak , 1 ≤ k ≤ ϑp} by p (the domain G, the visit to which we counted, forming the µ-th part of Zp ). 2
Remark 1. The next term of the development ln(1 + x) ≈ x − x2 + . . . (for small x) leads to the prediction, that one should expect the upper decline from the asymptotical value, N > ϑµ(p − 1).
Remark 2. Our reasoning (far from the mathematical rigour) might be considered as some kind of adiabatic replacement of the logarithmically nonuniform sequence (of the numbers ln(jp) = ln p + ln j and of the intervals ∆j , starting at these points) by the arithmetical progressions (of numbers and of intervals correspondingly) – see figure 5.1 at page 44. In fact the “step” ln(j + 1) − ln j = ln j+1 ∼ 1/j of the logarithmical j sequence is slightly declining when j is growing, and therefore this logarithmical sequence is not exactly an arithmetical progression (being, however, close to it for rather long intervals of change of j, provided that the step of the approximating arithmetical progression had been chosen correspondingly to the range of j). Were the logarithmical sequence an arithmetical progression, our reasoning would be based strictly by the Weyl fractional parts equidistribution theorem. Thus, the remaining foundation of our euristical reasoning depends on the evaluation of the error of the adiabatic approximation (which might also be replaced by the modification of the Weyl’s theorem proof, including the study of the behavior of the Fourier coefficients of the characteristical function of the union of the intervals ∆j ). For p = 997, ϑ = 1/2, µ = 1/2, A = 7, I had calculated the visits number N to be 279, which is larger than the conjectured asymptotical expression, ϑµ(p − 1) ≈ 249. For p = 1009, A = 11 (and ϑ = µ = 1/2) the visits number is N = 269 (for the visits of the residues 11k , 1 ≤ k ≤ 503, modulo 1009 to the domain
46
Adiabatic study of equipartition.
G = {1 ≤ x ≤ 504} (mod 1009)). The asymptotical expression provides ϑµ(p − 1) ≈ 252 (suggesting the decline of the difference between N/m and √ |G|/z like c/ p). It would be interesting to see more examples, for larger p, to evaluate the decline empirically. The quantities N (A)/m, corresponding to different primitive elements A, may deviate from the mean value for some exceptional values of A, and the evaluation of the dispersion of this deviation might provide an interesting information on the rarity of those exceptional values of A for large primes p. A different approach to the asymptotical equipartition of the sequence of the residues for the division by p of the members of the geometrical progression {Ak , 1 ≤ k ≤ ϑp} among all the residues of the division by p (A being a primitive residue, the number ϑ being fixed, 0 < ϑ < 1, and the prime number p tending to infinity) is provided by the following construction. Consider the multiplicative group of complex number Zp = {z ∈ C : z p = 1} . The functions on this group are the (Fourier) linear combinations of the characters, e0 ≡ 1,
e1 = z ,
e2 = z 2 , . . . ,
ep−1 = z p−1 .
The multiplication by A of the residues (of the division by p) can be represented in these notations as the mapping f : Zp → Zp , where f (x) = xA . This mapping acts on the functions as the linear operator (and algebra morphism) f ∗ ek = eAk . Function e0 is invariant under this mapping, while the remaining p − 1 characters are permuted cyclically (A being a primitive element). To prove the equipartition one have to prove the convergence to zero of the time averages of the functions, orthogonal to e0 . For the character ek we have to study the time average ebk =
T −1 X
(f ∗ )t ek /T .
t=0
To study this averaged function, make once more the Fourier transform, ordering the harmonics ek (k 6= 0) in the order of their numbers in the above
Chapter 5
47
cyclic permutation of order p − 1 (that is in the order of the sequence of the residues At (mod (p − 1)) in Zp−1 ). To do it we consider the multiplicative group of complex numbers, Zp−1 = {wt = e2πit/(p−1) } , where 0 ≤ t < p − 1. The corresponding harmonics E0 , . . . , Ep−2 : Zp−1 → C are defined by the formula Er (w) = w r . As it is explained above, we identify the characters ek (k 6= 0) with these functions Er in the way that the sequence e1 , f ∗ e1 , (f ∗ )2 e1 , . . . , (f ∗ )p−2 e1 takes the form E0 , E1 , E2 , . . . , Ep−2 , and therefore f ∗ takes the form Er → Er+1 of the multiplication by the function w(= E1 ). For the time average we obtain the expression T br = 1 (1 + w + w 2 + · · · + w T −1 )Er = 1 w − 1 Er , E T T w−1
tending to zero for large T (for every value of r and hence for all the characters ek , k 6= 0). Thus, the time average of any function on the group Zp−1 tends to its space average for T → ∞. The space average of the harmonic ek along Zp−1 is easy to compute: p−1
p−1
1 X 1 X e0 , (f ∗ )t ek = ek = − p − 1 t=0 p−1 p−1 k=1
P p since p−1 k=0 ek = 0 (by the Vieta theorem for the equation z = 1). Therefore, the mean value ebk tends to 0 for p → ∞ if k 6= 0. Thus the time average along the segment of geometrical progression {At } of any fixed linear combination of the harmonics on group Zp tends for p → ∞ to its space average. Applying this to the characteristic function of part G of group Zp , we would prove the asymptotical equipartition statement for the progression segment at the limit p → ∞.
48
Adiabatic study of equipartition.
This reasoning is, however, unsufficient for a rigorous proof, since the characteristic function of domain G is not a fixed linear combination of the harmonics: the number of the harmonical summands, needed to approximate this characteristic function of G on Zp is growing with p. Remark. In our study of the asymptotical equipartition for p → ∞ we had fixed the (Jordan measurable) domain G in the real continuous torus T n , evaluating the number N of visits of a sequence {Ak } to the corresponding domain G(p) of the finite n-torus Znp ; this domain consists of a finite number of points (growing with p). Perhaps, together with the natural domains G (similar to the band 0 ≤ u ≤ d in T 2 ), one might take also many more complicated domains G(p) (similar, say, to the set defined by the condition that u is even for the points (u, v) ∈ Znp ). The conjecture is that the equipartition asymptotics would hold for p → ∞ even for such “irregular domains”, provided that the algorithm, defining the domain G(p), would be sufficiently simply. I do not know any proved theorem of this kind (which would be interesting even for the distribution of the fractional parts of an arithmetical progression members on a circle), while one might consider as some confirmation of this conjecture the Skolem theorem on the zeroes of the recurrent sequences {ak }: this theorem claims, that the set {t : at = 0} consists of a finite set of arithmetical progressions of integers t, whatever be the recurrent sequence. The ergodic theory content of the conjectured chaoticity theorems is the statement, that a sufficient chaoticity of the dynamics obstructs the prediction of all those properties of the trajectories, which might be computed by simple algorithms. Remark. The study of the equipartition of the finite segments of geometrical progressions along the one-dimensional finite torus Zp “irregular” domains, described above, might be useful for the investigation of the degree of the equipartition of the geometrical progressions segments along finite fields with a large number z = pn of elements, living on an n-dimensional torus. The point is, that the mapping k 7→ Ak is sending bijectively the set of the nonzero residues (of the division by the number z − 1) on the set of the nonzero elements of the z elements field (living on an n-torus). Therefore, knowing the equipartition property for the arithmetical progression {tr, t = 1, 2, 3, . . . } on the 1-dimensional finite torus Zz−1 one might deduce the information on the partition of the geometrical progression {B t , t =
Chapter 5
49
1, 2, 3, . . . }, where B = Ar , along the finite n-torus Znp (consisting of z = pn points). For instance, to study the asymptotical behaviour of the number N of the visits of a segment of geometrical progression {B t } to domain G of the field of z elements, it would suffice to know the asymptotical behaviour of the number of the visits of the corresponding segment of arithmetical progression {tr, t = 1, 2, 3, . . . } to the full preimage f −1 G of domain G under the above bijection k 7→ Ak (which we have denoted here by f ). In this sense to prove the asymptotical equipartition of the segments {B t } of geometrical progressions along the n-dimensional (finite) torus it would suffice to prove the proportionality of the number of visits to subdomains of the finite circle Zz−1 by the arithmetic progression {tr} segment to the measure of the subdomain, but one needs to know this approximate proportionality inequality for (different from the ordinary intervals) subdomains of the finite circle, provided that the subdomain is defined by an algorithm of bounded complexity (like, say, the domain f −1 (G) for the domain G, that we are studying on the finite torus). While the arithmetical progressions are a much simpler object for equipartition study, than the geometrical progressions, the known results on the arithmetical progressions are unsufficient for two reasons: one needs to study the visits to complicated domains f −1 (G) and the one-dimensional finite torus Zz−1 points number z − 1 = pn − 1 is not prime. In spite of this formal inadequacy of the equipartition results on the finite circles to our problem, it seems that it might be not too difficult to attain the relevant arithmetical progressions equipartition results that are needed for the geometrical progressions equipartition studies along the finite n-tori.
50
Adiabatic study of equipartition.
Chapter 6 Projective structures, generated by a Galois field The Galois field’s algebra has a remarkable geometrical aspect, similar to the projective geometry aspect of linear algebra (including the ellipsoids and hyperboloids principal axis geometry instead of the quadratic forms eigenvalues theory). Calculations are usually simpler in the algebraic version, but the real understanding is reached only by the geometrical approach to the principal axis theory. Goethe told, that “mathematicians are similar to French people: they translate everything in their language, and it changes all the content”. I shall describe now the geometrical version of the Galois field algebra: the theory of the projective structures on finite sets and the study of the action on them of the groups of the “Frobenius transformations” (of powers calculations) Φk (x) = xk . Recall first the projective line notation. Consider all the straight lines, containing point 0, on the usual plane R2 . The manifold of all such lines is one-dimensional, and it is diffeomorphic to the circle. To see it, describe the points of the plane by their Cartesian coordinates (u, v). The line is then described by its equation, u = λv, where λ is a constant (depending on the chosen line and defining it) – see figure 6.1. The constant λ is called the affine coordinate on the projective line (whose 51
52
Projective study of Galois table. u u = λv λ
0
1
v
Figure 6.1: A projective line and its affine coordinate λ. points are the straight lines of the plane, going through the origin 0). However, like the whole Earth sphere is not represented on one hemisphere map, the affine coordinate λ values do not provide all the straight lines, going through the origin. Namely, they do not provide the vertical line (v = 0 at figure 6.1). Therefore one adds an “infinite” point λ = ∞ to the axis of the variable λ, providing the description of the whole real projective line RP 1 by the values RP 1 ∼ {λ ∈ R, λ = ∞} .
This “infinite point” λ = ∞ of the projective line (representing the vertical line v = 0 of figure 6.1) is as good as all the others, since the vertical line on the plane is as good as the others (for instance, they are transformed to any of them by the plane rotations). A different choice of the initial coordinates system on the plane (say, one ˜ might take u˜ = v, v˜ = u) would produce a different affine coordinate λ ˜ ˜ (λ = u ˜/˜ v in our example) and a different point (λ = ∞) on the projective 1 line RP (which would correspond in our example to the horizontal line, v˜ = 0, of figure 6.1). ˜ = 0 in our In the neighbourhood of the vertical line of figure 6.1, (where λ ˜ (λ ˜ = 1/λ in our example) is regularly example) the new affine coordinate λ parametrizing the real projective line. Therefore, the tending of the affine ˜ = 0) of the manifold of coordinate λ to +∞ or −∞ leads to the same place (λ the straight lines of the plane, containing the origin. This compact manifold, the real projective line, is therefore diffeomorphic to the circle (figure 6.2): RP 1 ≈ S 1 .
Chapter 6
53 ˜=0 λ
λ → −∞
˜ λ
λ → +∞
λ λ=0
Figure 6.2: The real projective line diffeomorphism to a circle. For a different linear coordinates choice on the plane (which might be ˜ would be a fractionally-linear nonorthogonal) the new affine coordinate λ function of the old one, ˜ = aλ + b , (6.1) λ cλ + d since the new coordinates have the form u ˜ = au + bv ,
v˜ = cu + dv .
The new coordinate axis should not coincide, therefore ad 6= bc. The transformation of the axis of λ, defined by the formula (6.1) is called a projective transformation. The axis of λ is considered here as being completed by the infinite point, and the formula (6.1) defines a diffeomorphism of the real projective line (that is of a circle) to itself. Of course, the algebraic versions of this simple geometry are the conventions ˜ = ∞ for cλ + d = 0 , λ ˜ = a/c for λ = ∞ . λ The real projective space of dimension n − 1,
RP n−1 = (Rn r 0)/(R r 0) , is defined similarly to the projective line, being the manifold of the straight lines, going through the origin 0 of the n-dimensional vector space Rn . Its affine chart is constructed from a linear coordinates system (u1 , . . . , un ) in Rn : if un 6= 0, we define the vector λ ∈ Rn−1 , whose coordinates are λ1 = u1 /un , . . . , λn−1 = un−1 /un .
54
Projective study of Galois table.
In other terms, we take the intersection point λ of the line that we wish to describe with the hyperplane un = 1 of space Rn , this point λ is the image of the line on the affine chart Rn−1 (similar to the situation of figure 6.1 at page 52, where n = 2). To obtain all the straight lines of space Rn , going through the origin, one has to use n such affine charts (represented by the n hyperplanes, {un = ˜ at figure 0}, . . . , {u1 = 0}. Thus, for n = 2 one needs two charts (λ and λ 6.2 of page 53). The corresponding fractionally-linear transformations are described in RP m by the following extension of formula (6.1): for j = 1, . . . , m the coordinates of the image point of the point λ with affine coordinates (λ1 , . . . , λm ) are ˜ j = aj,1 λ1 + · · · + aj,m λm λ b 1 λ1 + · · · + b m λm
(it is important to observe the denominator’s independence on the number j ˜ j ). of coordinate λ The geometrical meaning of these algebraic formulae is that they describe the projective transformations which are obtained (say, for m = 2) when one projects one plane P in the 3-space onto another plane P˜ by the rays, starting at a common projection center (figure 6.3). 0
P
P˜
Figure 6.3: A projective transformation of a cat. Therefore, the theory that we are describing is basic both for the geometry
Chapter 6
55
of the projections, sending the straight lines to the straight lines, and to the theory of the perspective (where the rails of a straight railway “meet at an infinitely far point on the horizon”). The great Italian painter Paolo Uccello (whose name means “Bird”), being one of the first painters, seriously studying the mathematical theory of the correct perspective drawing, answered to his wife, inviting him at midnight to the bedroom “I am coming – what a nice perspective”, having in mind the beauty of his remarkable drawing. To become familiar with this projective geometry, one might try to prove the following facts: 1. The real projective plane is unorientable, the real projective spaces RP m being orientable for odd dimensions m and nonorientable for the even dimensions. 2. The complement to a small disc on the real projective plane is diffeomorphic to the M¨ obius band (which surface had been discovered by M¨obius, due to this fact). The complex projective spaces CP m are defined similarly to the real one, starting from the complex vector space Cn (where n is still equal to m + 1). The points of this complex projective space are the complex straight lines, going through the origin of space Cn : CP n−1 = (Cn r 0)/(C r 0) . The affine coordinates and the projective transformations are defined by the same formulae, as in the real case1 . In the complex projective spaces and transformations cases the difficulty, described above, vanishes. Thus, the complex projective line, CP 1 , is obtainable from the axis of the complex variable λ by the addition of one infinite point. The resulting variety is diffeomorphic to the ordinary sphere S 2 and is called the Riemann sphere. In the neighbourhood of the “infinite” point λ = ∞ the affine complex ˜ = 1/λ. coordinate is the function λ 1
This is the important advantage of algebra: the algebraists are ready to apply their formulae to the objects, which are quite different from those, for which these formulae had been proved, and if the result occurs to be wrong, they postulate it to be presumably true for their “ideal” objects, replacing the difficult reality study by the easier investigations of these “ideal” objects.
56
Projective study of Galois table.
The complex straight lines, going through the origin 0 of the complex space Cn , intersect the sphere, centered at the origin (which is defined by P the equation |uk |2 = 1 and is diffeomorphic to the sphere S 2n−1 of the real vector space R2n ) along the real circles S 1 . n The real lines S m (where P in 2R , going through zero intersect the sphere m = n − 1, |uk | = 1) along 0-dimensional spheres S 0 (each of which consists of two opposite points). While the real projective space RP m might be obtained from the sphere S m by the gluings of all pairs of opposite points, which are proportional vectors of the real vector space, RP m = S m /(S 0 = {±1}) , the complex projective space manifold, CP m might be obtained from the sphere S 2n−1 (where n = m + 1), gluing at one point each circle S 1 , along which the sphere intersects a complex straight line, going through the origin. All the points of this circle are complex proportional vectors of the complex vector space, and they are obtainable from (arbitrary) one of the points of this circle, multiplying it by all the complex number of modulus one: CP m = S 2m+1 /(S 1 = {eiϕ }) . The really 4-dimensional manifold CP 2 is obtainable from the affine complex plane C2 by the addition “at the infinity” of a complex projective line CP 1 , that is of a Riemann sphere: CP 2 = C2 ∪ S 2 . The complex projective line CP 1 might also be described as the manifold, whose points are the special great circle S 1 ⊂ S 3 . A special great circle is the intersection of the 3-sphere S 3 with a complex straight line in C2 , containing the origin 0. This circle is the union of the points of sphere S 3 which are complex proportional: CP 1 = S 3 /(S 1 = {eiϕ }) . It is interesting to notice, that different special great circles in the 3-dimensional sphere S 3 , which is fibred into these circles, are situated in the sphere specially: the linking coefficient of any pair of such special great circles is equal to one.
Chapter 6
57
The linking coefficient of two disjoint oriented smooth closed curves in the oriented 3-sphere (or in the oriented Euclidean 3-space) is defined as the intersection index of one of these circles with the smooth oriented immersed compact surface, whose boundary is the other circle (figure 6.4).
Figure 6.4: Two curves, having the linking coefficient equal to two. The orientations are used here the following way: the orienting frame of the surface at a boundary point consists of the orienting vector of the boundary curve, followed by the tangent vector to the surface, directed inside it. The intersection index of the second curve with the surface, whose bound is the first curve, is counting the intersection points of these two objects, equipped with the signs, which are positive, if the 3-frame, formed by the three vectors, orienting the curve and the surface, orients positively the 3space. The linking coefficient of two curves does not depend on the choice of the surface, bounded by one of the curves: the surface should only be nowhere tangent to the second curve. Linking coefficient L of two oriented curves is symmetric with respect to the curves ordering: L(I, II) = L(II, I). The 3-sphere fibration, whose fibers are the above special great circles, is called “Hopf fibration” S 3 → (CP 1 = S 2 ), whose fiber is S 1 . This fibration is the basic object for many branches of mathematics. The 3-sphere is the ordinary Euclidean 3-space, compactified by the addition of one point. In this model of the 3-sphere S 3 , the Hopf fibration becomes the decomposition of the Euclidean 3-space into a straight line (originated from the circle, containing the added point) and the complement to this line, fibered into closed curves, whose pairwise linking coefficients are all equal to
58
Projective study of Galois table.
one. While I am able to draw the resulting nice picture, I shall not show it in this lecture, leaving to the listener the pleasure to draw it themselves. Instead we shall transfer now the theory, which we had described above in the real and in the complex case, to the case where the numbers are replaced by the residues of the division by a prime number p: we shall define the finite projective spaces P m (Zp ), very similar to the real manifolds RP m and to the complex manifolds CP m (but consisting of finite numbers of points). We shall start from the finite projective line, P 1 (Zp ). It is defined as the set of the straight lines of finite plane Z2p , containing the origin: P 1 (Zp ) = (Z2p r 0)/(Zp r 0) . In terms of the coordinates (u, v) on the finite plane (which are now residues modulo p) the straight line equation has the form u = λv, but the “affine coordinate” λ ∈ Zp is also a residue modulo p. To attain all the straight lines, one should add to these p values of the affine coordinate λ one more value, denoted by the symbol ∞ (to include the vertical line v = 0, taking into account that λ = u/v for v different from 0). Therefore, the finite projective line P 1 (Zp ) consists of p + 1 points: |P 1 (Zp )| = p + 1 (λ = 1, 2, . . . , p; ∞) . The projective transformations λ 7→
aλ + b , cλ + d
(6.2)
(where a, b, c and d belong to Zp , ad − bc being different from zero) permute some way the p+1 points of the finite projective line, but these permutations are not arbitrary. Indeed, the symmetric group S(p + 1) of all the permutations of p + 1 points of our finite projective line consists of (p + 1)! permutations, and the order of the group of the projective transformations (6.2) is much smaller. Lemma. The group PL(Zp ) of the projective transformations ( 6.2) of the finite projective line, consisting of p + 1 points, is formed by p(p2 − 1) permutations.
Chapter 6
59
Indeed, if a 6= 0, we might divide all the coefficients by a, writing the transformation (6.2) in a form, where a ˜ = 1 The other coefficients, ˜b and c˜, may have (independently of each other) p values each, and the remaining ˜ should verify the condition d˜ 6= ˜b˜ coefficient, d, c, having (for fixed values of ˜b and of c˜) p − 1 possible values. This way we get p2 (p − 1) transformations (6.2). In the remaining case a = 0 the nondegeneracy condition would be bc 6= 0. If d 6= 0, we might, dividing by d, reduce the formula (6.2) to the form, where d˜ = 1; the number of such transformations is (p − 1)2 , since ˜b˜ c 6= 0. Finally, in the case a = d = 0, the transformation has the form λ 7→ bλ, such transformations number being p − 1 (since b 6= 0). Thus, the total number of the projective transformations of the finite projective line P 1 (Zp ) of all the three kinds is equal to the sum p2 (p − 1) + (p − 1)2 + (p − 1) = (p − 1)(p2 + (p − 1) + 1) = p(p2 − 1) , proving the Lemma. The factorial, (p + 1)!, is much larger, if p is large (starting already at p = 5, where (p + 1)! = 720, p(p2 − 1) = 120). The point is, that the subgroup of the projective permutations of the points of the finite projective line forms in the whole permutation group S(p + 1) a small subset of those permutations which preserve some remarkable geometrical structure in the finite set P 1 (Zp ), consisting of p + 1 points. Unfortunately, I have no geometrical description of this remarkable finite projective line structure, while algebraically the structure is described by an affine coordinate λ ∈ Zp on the complement to some “infinitely far situated” point λ = ∞ of the finite set M , equipped with the projective structure. ˜ : (M r •) → Zp defines the same proA different similar coordinate, λ jective structure on M , as the affine coordinate λ, if it is related to λ by a projective transformation (6.2). While this algebraic description of the projective structures on a finite set M is rather nongeometric, it will be used below for the study of those projective structures on the sets of p + 1 elements, which are generated by the Galois fields of p2 elements. We shall also use it to study the Frobenius mappings actions on these finite sets’ projective structures. Namely, fixing a multiplicative generator A for the field pf p2 elements, we identify bijectively this field with the finite plane (or torus) Z2p of the field’s table, as it is described above (in §2).
60
Projective study of Galois table.
Consider now the finite projective line P 1 (Zp ), whose p + 1 points are the p + 1 straight lines of the finite plane Z2p containing the origin. Lemma. The set of the straight lines, containing the origin, of the finite plane Z2p , considered as the field of p2 elements, does not depend on the choice of the multiplicative generator A, which had been used to identify the field with the finite plane. Proof. For two proportional points x and cx for a scalar c = 1 + · · · + 1 (on the same line in the table), the corresponding field’s elements are also scalar proportional: Ak and Ak + · · · + Ak = cAk . Since this scalar proportionality relation does not depend on the choice of the generator A, the Lemma is proved. Differently from the set of lines, its projective structure depends in general from the choosen generator: different choices of the multiplicative generators (providing different tables of the same field) are generating different projective structures on the same intrinsically defined set of p + 1 straight lines. We shall study the examples of such structures in the next paragraph. Remark. The above constructions are easily adapted to the Galois fields with pn elements (for every n). The straight lines, containing the origin, form in this case a finite set M , whose points number equals to |P m (Zp )| =
pn − 1 = pm + pm−1 + · · · + 1 p−1
(where n = m + 1). The field’s table (depending on the choice of a multiplicative generator A) defines a finite projective structure of the space P m (Zp ) on finite set M of the lines. However, unlike the set-theoretical structure of M , this projective structure is not intrinsic, depending on the generator A choice, used for the identification of the field with the finite torus. Therefore, the projective geometry of the Galois field should study several projective structures on the same finite set M (whose number equals, as we shall see soon, the value of the Euler function, ϕ(z − 1) for the field of z = pn elements). The Euler’s function ϕ value ϕ(x) is, by definition, the number of those residues (of the division by natural number x), which are relatively prime
Chapter 6
61
to x. For instance, for a prime number p one has ϕ(p) = p − 1, ϕ(pn ) = (p − 1)pn−1 , and for mutually prime arguments x and y, Euler function is multiplicative: ϕ(x, y) = ϕ(x) · ϕ(y) . Thus, ϕ(24) = ϕ(3)ϕ(8) = 8 , ϕ(48) = ϕ(3)ϕ(16) = 16 , ϕ(120) = ϕ(8)ϕ(3)ϕ(5) = 32 , ϕ(168) = ϕ(8)ϕ(3)ϕ(7) = 48 .
62
Projective study of Galois table.
Chapter 7 Calculation of projective structures of finite projective lines, generated by fields of p2 elements, and of Frobenius transformations group action on these lines and on these structures Consider as the simplest example the field of 25 elements (that is, the case p = 5). This field’s table is calculated above (see page 17) for the multiplicative generator, corresponding to the matrix (A) = ( 01 22 ). The finite projective line P 1 (Z5 ), corresponding to this field and to this table, consists of p + 1 = 6 points defined by the values of the affine coordinates λk = uk /vk (for the element Ak of the field). Thus, the table of page 17 provides the following 6 straight lines containing the origin: 63
64
Frobenius transformations actions. λ k k mod 6 0 24, 18, 6, 12 0 1 8, 2, 14, 20 2 2 11, 5, 17, 23 5 3 10, 4, 16, 22 4 4 15, 9, 21, 3 3 ∞ 1, 19, 7, 13 1
The last column of this table is very useful, simplifying many succeeding calculations, and we shall prove it now in a more general form. Lemma. The straight lines, containing the origin 0 of the plane Z2p , represent those sets {Ak } of field’s elements, for which k = const (mod (p + 1)) (for the k corresponding to the nonzero elements of the field). This congruence remains true for any choice of the multiplicative generator A, defining the table Z2p of the field, consisting of p2 elements. Proof. The condition Ak = cA` , where c belongs to the scalars, which defines the belonging of k and ` to the same line, can be written in the form << Ak−` = c is a scalar element >> . The scalar subgroup in the multiplicative group {Ak }, having p2 −1 elements, consists of the p − 1 elements, {c = 1, c = 2, . . . , c = p − 1}. The degrees s, corresponding to the elements of this subgroup, {c = As } form an arithmetical progression of p − 1 terms in the additive group Zz−1 , z = p2 . Therefore, the step of the arithmetical progression equals to (p2 − 1)/(p − 1) = p + 1. Hence, this progression has the form {s = (p + 1)r}, 2 where r ∈ {1, 2, . . . , p − 1} (since the element Ap −1 = 1 belongs to the subgroup of the scalars {c} and hence the point s = p2 − 1 belongs to the arithmetical progression {s}). Therefore, the necessary and sufficient condition for the belonging of the nonzero elements Ak and A` of the field to the same line is the relation k − ` = (p + 1)r, where r ∈ Zp−1 , proving the Lemma. The multiplicative generators of the field of p2 = 25 elements are As , where s is relatively prime to p2 − 1 = 24. There are ϕ(25 − 1) = 8 of them: s ∈ {1, 5, 7, 11, 13, 17, 19, 23} .
Chapter 7
65
To see the permutation of the 6 points of the projective line, produced by the replacement of the generator A by the generator As = As , it suffices to take one point k, for which λ attains a choosen value, and to represent k −ks Ak in terms of As . Since A = A−s s , we obtain A = As . Therefore, the new choice of the generator acts on each line the same way, as the Frobenius transformation Φ−s = Φ−1 s . For this reason we shall study now the action of the Frobenius transformations Φs on our straight lines. To do it, let us calculate the affine coordinate of the image of the straight line, whose preimage affine coordinate is λ, and denote it λs (λ1 ) = λ(Φs (x)) , for λ(x) = λ1 . Calculating function λs of λ1 by these formulae, we may use that the relation x = Ak implies Φs (x) = Aks , and therefore to calculate λs (λ1 ) it suffices to associate to λ1 any k, provided by the preceding table, to multiply it by the number s and to find the value of function λ at this product ks, provided by the same table (used this time in the opposite direction). The value λ(Ak ) depending only of the residue k (mod 6) of the division of k by 6, it suffices to multiply by s just this residue (rather than k). This leads in few minutes to the following table of the values of all the 8 functions λs on all the 6 lines: λ1 λ5 λ7 λ11 λ13 λ17 λ19 λ23
0 0 0 0 0 0 0 0
1 3 1 3 1 3 1 3
2 ∞ 2 ∞ 2 ∞ 2 ∞
3 1 3 1 3 1 3 1
4 4 4 4 4 4 4 4
∞ 2 ∞ 2 ∞ 2 ∞ 2
One might abbreviate these calculations even more, taking into account the congruences 1 ≡ 7 ≡ 13 ≡ 19 (mod 6) , implying the congruences
λ1 ≡ λ7 ≡ λ13 ≡ λ19 , and hence the identities for the actions P Φs of Φs on the projective line, P Φ7 = P Φ13 = P Φ19 = id ,
66
Frobenius transformations actions.
(id being the identical transformation of P 1 (Z5 ), leaving fixed all its 6 points). Similarly, we observe the congruences 5 ≡ 11 ≡ 17 ≡ 23 (mod 6) , whence λ5 ≡ λ11 ≡ λ17 ≡ λ23 , and therefore the corresponding Frobenius transforms are acting on the projective line the same way: P Φ5 = P Φ11 = P Φ17 = P Φ23 . Thus we have computed the homomorphism ψ of the Euler group Γ (formed by the transformations Φs or by those residues s of the division by p2 − 1 = 24, which are relatively prime to the number p2 − 1) onto its projectivized version: ψ : {Φs } −→ {P Φs } , P Φs ∈ S(p + 1) being the permutation of the p + 1 straight lines, containing the origin, by the Frobenius transformation Φs of the field, consisting of p2 elements. For the case p = 5 we had calculated the answers: 1) Γ ≈ Z32 (generated by a = 5, b = 7, c = 13, verifying the identities 11 = ab, 17 = ac, 23 = abc); 2) ψ(Γ) ≈ Z2 (whose nontrivial element P Φ5 acts on the λ axis as the reflection of the diagram 1 2 0 4 | | ; 3 ∞ in the horizontal mirror). 3) To check, whether the permutation P Φ5 of 6 points is projective, note that 0 7→ 0 means that λ5 is zero when λ1 = 0. Therefore, be the permutation projective, it should have the form λ5 = aλ1 /(cλ1 + d) .
Chapter 7
67
The property 2 7→ ∞ of P Φ5 implies, that λ5 should be ∞ for λ1 = 2, and therefore one should have 2c + d = 0. Similarly, ∞ 7→ 2 means, that one should have λ5 = 2 for λ1 = ∞, and therefore one should have a = 2c. We obtain therefore, that d = −2c, a = 2c, providing for the projective transformation P Φ5 the form λ5 = 2λ1 /(λ1 − 2) . This is, indeed, true for all the 6 values of λ1 . Thus, the final conclusions are: 1) The projective structure on the line P 1 (Z5 ) does not depend on the choice of the multiplicative generator, used to identify the field to the finite torus: all the 8 choices lead to the same projective structure. 2) The kernel of the projectivization homomorphism ψ consists of the 4 Frobenius transformations {Φ1 , Φ7 , Φ13 , Φ19 }, which move no straight line, containing 0. This kernel forms a group, isomorphic to Z22 . 3) The image of the projectivization homomorphism ψ is isomorphic to Z 2 . Its only nontrivial permutation of the 6 points of the finite projective line P 1 (Z5 ) does not move the points λ1 = 0 and λ1 = 4, permuting λ1 = 1 with λ1 = 3 and λ1 = 2 with λ1 = ∞. This projective transformation is defined by the formula λ5 =
2λ1 ; λ1 − 2
it is generated by the automorphism Φ5 of the field of 25 elements (verifying the identity Φ5 (x + y) = Φ5 (x) + Φ5 (y) , which is not verified by Φ7 ). The calculations of the projective structures and of the Frobenius mappings (which might be or not be Frobenius automorphisms) for the fields, consisting of p2 elements with different prime numbers p, follows the same lines, as in the case p = 5, studied above. But the resulting answers are so different for different prime numbers (p = 7, 11, 13), that I have even no guesses on the general answers for the higher values of p.
68
Frobenius transformations actions.
Case p = 7. Euler group Γ(48) consists of ϕ(p2 − 1) = ϕ(48) = ϕ(3) ϕ(16) = 16 residues modulo 48, relatively prime to 48: {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47} . This multiplicative group is isomorphic to direct product Z4 × Z2 × Z2 , with generators 5 for Z4 and 7, 17 for Z2 factors. The 8 points of finite projective line P 1 (Z7 ), corresponding to elements Ak of the field, where (A) = ( 01 22 ), are defined by the corresponding residues (of the division of exponents k by number p + 1 = 8) the following way: λ1 (Ak ) k (mod 8)
0 1 0 2
2 3 6 7
4 5 6 5 3 4
∞ 1
(see the table of the field at page 19). The actions of the generators of Euler group, represented by the Frobenius mappings Φ5 , Φ7 and Φ17 , provide the following values of functions λ5 , λ7 , λ17 (calculated, multiplying k by s, by an algorithm, described in details above, for the case p = 5): λk (λ) = λ1 (Φk (x)), for λ1 (x) = λ. λ1 λ5 λ7 λ17
0 0 0 0
1 1 2 1
2 2 1 2
3 4 5 5 ∞ 3 ∞ 5 4 3 4 5
6 ∞ 6 4 6 3 6 ∞
We need no new calculations for P Φ17 , since 17 ≡ 1 (mod 8), and therefore permutation P Φ17 = P Φ1 = id is the identical transformation of finite projective line P 1 (Z7 ) (leaving each point of this line fixed, λ17 ≡ λ1 ). Permutations P Φ5 and P Φ7 act on the points, corresponding to the different values of λ, like act the mirror symmetries (with respect to the horizontal mirror) of the following 2 diagrams, whose points represent the points of P 1 , denoted by their coordinate λ1 : Case P Φ5 : 3 4 | | 0 1 2 6; 5 ∞
Chapter 7
69
Case P Φ7 : 1 3 4 | | | 0 6. 2 ∞ 5 The first permutation (P Φ5 ) is not projective, having 4 fixed points (such a projective mapping ought to be identical, preserving all the 8 points). For the second permutation (P Φ7 ) we deduce from 0 7→ 0, that in the 1 , from 3 7→ ∞ it follows then, that 3c+d = 0, while projective case λ7 = cλaλ1 +d ∞ 7→ 3 implies the relation a = 3c. Thus, in the projective case the values of λ5 ought to be everywhere equal to 3λ1 /(λ1 − 3), and it is indeed the case in the table of page 68. Thus permutation P Φ7 is projective (preserves the projective structure). Multiplying the permutations, that we had already calculated, we obtain the whole projectivization homomorphism ψ : (Γ ≈ {Φs }) → {P Φs }. The resulting conclusions are: 1) The field, consisting of 49 elements, generates 2 different projective structures, defined by functions λ = λ(k) on the coordinates k (mod 8) of the 8 points of projective line P 1 (Z7 ) (correspondingly to the different choices of the multiplicative group’s generator). Permutation P Φ 5 sends one of these two structures to the other. The difference between these structures is similar to the difference between the old and new railways schedules, due to the cities renaming. 2) Permutation P Φ7 preserves both projective structures of 8 elements set {k (mod 8)}. 3) Permutation P Φ17 = P Φ1 is identical, preserving every point of set P 1 (Z7 ) (Frobenius mapping Φ17 belongs to the kernel of the projectivization homomorphism ψ). This kernel consists of the 4 Frobenius mappings Φs , Ker ψ = {Φ1 , Φ17 , Φ25 , Φ41 } , for which s is congruent to 1 modulo 8. This group is isomorphic to group Z22 (for instance, Φ17 Φ25 = Φ41 ). 4) The image of projectivization homomorphism ψ is also isomorphic to group Z22 . It consists of the four permutations {P Φ1 , P Φ5 , P Φ7 , P Φ11 }, of which (P Φ1 = id and P Φ7 ) are biprojective (preserving each of both
70
Frobenius transformations actions. projective structures), while each permutation P Φ5 and P Φ11 permutes the two projective structures of set P 1 (Z7 ). Frobenius transformation Φ7 is an automorphism of the field consisting of 49 elements, Φ7 (x + y) = Φ7 (x) + Φ7 (y) (which is not the case for Φ5 or for Φ11 ). Thus, group {P Φs } ≈ Z22 acts on the two projective structures of set P 1 (Z7 ), as Z2 (with kernel {P Φ1 , P Φ7 }, generated by the Frobenius automorphisms of the field). To obtain the first or the second projective structure defined by the coordinates k mod 8 of the function λ = λk of the 8 elements set P 1 (Z7 ), one has to choose the following multiplicative group’s generators As (as it is implied by the field’s table, page 19): P1 P5
1, 7, 17, 23 5, 11, 13, 19
25, 31, 41, 47 29, 35, 37, 43
s = 8r ± 1 s = 8r ± 3
Frobenius trabsformations of the first line (where s = 8r ± 1) preserve both projective structures P1 and P5 , while that of the second (where s = 8r ± 3) permute projective structures P1 and P5 of set P 1 (Z7 ).
Case p = 11. Euler group Γ(120) consists of ϕ(p2 − 1) = ϕ(120) = ϕ(3) ϕ(5) ϕ(8) = 32 residues modulo 120, relatively prime to 120: they are the residues {1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, . . . } , (including 120 − s together with s). This multiplicative group is isomorphic to the direct product Z4 × Z32 (with generators 7 for Z4 and 11, 19, 61 for the 3 factors Z2 ). The 12 points of projective line P 1 (Z11 ), corresponding to elements Ak of the field (matrix (A) being ( 01 31 )) depend on the residues of the division of the numbers k by p + 1 = 12 the following way: λ1 (Ak ) k (mod 12)
0 1 2 3 4 5 6 7 8 9 10 ∞ 0 9 7 5 2 3 8 4 10 6 11 1
Chapter 7
71
(it follows from the field’s table, page 19). The actions of the generators of Euler group by Frobenius mappings Φ7 , Φ11 , Φ19 and Φ61 provide the following table of the values of functions λ7 , λ11 , λ19 , λ61 on the coordinates k (mod 12) of the points of set P 1 (defining the permutations P Φ7 , . . . , P Φ61 of the 12 points, forming this set P 1 (Z11 )): λs (λ1 (x)) := λ1 (Φs (x)) . As it is explained above (for p = 5), the calculation of λs (λ1 ) from table of page 70 follows the algorithm (λ1 7→ k) , (k 7→ sk) , (sk 7→ λ1 (Ask )) , where we first use page 70 table the downstairs (↓) way and at the end the opposite, upstairs way (↑). These calculations have to be performed only for one representative of each of the 4 modulo 12 classes of the numbers s ∈ Γ: Γ1 Γ7 Γ11 Γ17
1 ∼ 13 ∼ 37 ∼ 49 ∼ 61 ∼ 73 ∼ 97 ∼ 109 7 ∼ 19 ∼ 31 ∼ 43 ∼ 67 ∼ 79 ∼ 91 ∼ 103 11 ∼ 23 ∼ 47 ∼ 59 ∼ 71 ∼ 83 ∼ 107 ∼ 119 17 ∼ 29 ∼ 41 ∼ 53 ∼ 77 ∼ 89 ∼ 101 ∼ 113
In case where s belongs to Γ1 mapping P Φs is identical, hence λ1 = λ13 = λ37 = · · · = λ109 . To calculate permutation P Φ17 (and functions λ17 = λ29 = · · · = λ113 ), it suffices to multiply permutations P Φ7 and P Φ11 (since 7 · 11 = 77 ∈ Γ17 ). It remains therefore to calculate functions λ7 and λ11 , for which the values are provided by the page 70 table of the values of λ1 (Ak ) (using the algorithm λ1 7→ k 7→ sk 7→ λ1 (Ask )): λ1 λ7 λ11 λ17
0 0 0 0
1 2 5 ∞ 5 3 1 10
3 10 2 ∞
4 4 8 8
5 1 1 5
6 6 7 7
7 7 6 6
8 8 4 4
9 10 9 3 9 ∞ 9 2
∞ 2 10 3
Permutation P Φ7 , P Φ11 and P Φ17 permute the 12 points of finite projective line P 1 (Z11 ) set. Denoting the points of this set by the values of coordinate λ1 , we can describe these permutations as the mirror symmetries of the following three diagrams (in the horizontal mirrors):
72
Frobenius transformations actions.
Case P Φ7 : 1 2 3 | | | 0 4 6 7 8 9, 5 ∞ 10 Case P Φ11 : 1 2 4 6 10 | | | | | 0 9, 5 3 8 7 ∞ Case P Φ17 : 2 3 4 6 | | | | 0 1 5 9. 10 ∞ 8 7
These diagrams imply, that involutions P Φ7 and P Φ17 do not preserve the projective structure P1 of finite line P 1 (Z11 ) (which is defined by coordinate 1 being a fractionallyλ1 ), while involution P Φ11 preserves it (λ11 = − λ1λ+1 linear function). Both permutations P Φ7 and P Φ17 are sending projective structure P1 to the same image projective structure: P7 := (P Φ7 )(P1 ) = P17 := (P Φ17 (P1 )) , functions λ7 and λ17 being related fractionally-linearly: λ17 = −
λ7 . λ7 + 1
Permutation P Φ11 (generated by Frobenius automorphism Φ11 of the field, consisting of 121 elements) preserves both projective structures, P1 and (P7 = P11 ) of set P 1 (Z11 ), consisting of 12 points, while each of the two permutations P Φ7 and P Φ17 permutes these two structures, P1 and P7 . The kernel of the projectivization homomorphism ψ : (Γ ≈ {Φs }) → {P Φs }, consists of the 8 Frobenius mappings Φs , forming Γ1 , where s = 12r + 1, that is where s ∈ {1, 13, 37, 49, 61, 73, 97, 109}. These 8 mappings form the group Ker ψ ≈ Z4 × Z2 (whose generators correspond to s = 13 for Z4 and to s = 61 for Z2 ). The image of projectivization homomorphism ψ consists of the 4 permutations {P Φ1 , P Φ7 , P Φ11 , P Φ17 }, forming a group, isomorphic to Z22 . Its
Chapter 7
73
action on the set of points of projective line P 1 (Z11 ) and on its two projective structures P1 and P7 (provided by the coordinates functions λ = λ(k), defined by the field of 121 elements for different choices of multiplicative group’s generator) is described above (in particularly, P Φ17 = (P Φ11 )(P Φ7 ), (P Φ11 )(P Φ11 ) = (P Φ7 )(P Φ7 ) = 1). All these facts mean, that Euler group Γ, represented by 32 Frobenius mappings, acts on the set of two projective structures P1 and P7 on the coordinates k (mod 12) of points of the finite projective line (generated by the field, consisting of 121 elements) as the group Z2 of the permutations of these two structures. Both structures remain fixed under the 16 mappings P Φs , for which s = 12r ± 1 (r ∈ Z) (that is for s, belonging to the above lists Γ1 and Γ11 ). The 16 permutations P Φs , for which s = 12r ± 5 (r ∈ Z) (that is those, for which s belongs to above lists Γ7 and Γ17 ) are sending P1 to P7 and P7 to P1 . Thus, the kernel of homomorphism ϕ : Γ → Z2 , defined by the actions of the Frobenius mappings on the projective structures, is Ker ϕ = Γ1 ∪ Γ11 ≈ Z4 × Z22 (with generators, corresponding to Φ13 for Z4 and to Φ61 and Φ11 for Z2 factors).
Case p = 13. Euler group Γ consists of ϕ(p2 − 1) = ϕ(168) = ϕ(3) ϕ(7) ϕ(8) = 48 residues modulo 168, relatively prime to the number 168: namely of residues {1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, . . . } , (including to Γ residue 168 − s together with residue s). This group is isomorphic to the product of 4 cyclic groups, Z6 × Z32 (generators: 5 for Z6 , 29, 43 and 85 for the second order factors). The 14 points of projective line P 1 (Z13 ), corresponding to elements Ak of the 169 elements field and to choosen generator (A) = ( 01 24 ), are defined by the residues of the division of k by p + 1 = 14 the following way: λ1 (Ak ) k (mod 14)
0 1 2 3 0 8 2 13
4 5 6 7 8 9 10 11 12 ∞ 11 6 7 12 4 9 10 5 3 1
74
Frobenius transformations actions.
(as the table at page 20 implies). The actions of Euler group’s generators by Frobenius transformations {Φ5 , Φ29 , Φ43 , Φ85 } provide the values of functions λ5 , λ29 , λ43 and λ85 , using the usual algorithm (λ1 7→ k) , (k 7→ sk) , (λs = λ1 (Ask )) . It follows, that the resulting permutation P Φs depends only on the residue of the division of s by p+1 = 14, and therefore the Euler group Γ is subdivided into 6 classes of residues s (mod 168), equiresidual modulo 14:
Γ1 Γ5 Γ11 Γ13 Γ17 Γ23
= {1, 29, 43, 71, 85, 113, 127, 155} , = {5, 19, 47, 61, 89, 103, 131, 145} , = {11, 25, 53, 67, 95, 109, 137, 151} , = {13, 41, 55, 83, 97, 125, 139, 167} , = {17, 31, 59, 73, 101, 115, 143, 157} , = {23, 37, 65, 79, 107, 121, 149, 163} ,
s = 14r + 1 s = 14r + 5 s = 14r − 3 s = 14r − 1 s = 14r + 3 s = 14r − 5
; ; ; ; ; ;
The table at page 73, relating k to λ1 (Ak ), provides the following values of functions λs : λ1 λ5 λ11 λ13 λ17 λ23
0 1 2 3 0 7 10 9 0 8 1 12 0 5 7 ∞ 0 10 5 4 0 2 8 11
4 5 6 3 2 6 9 10 6 12 1 6 11 8 6 ∞ 7 6
7 8 9 10 11 8 5 12 1 4 5 2 ∞ 7 3 2 10 11 8 9 1 7 3 2 ∞ 10 1 4 5 12
12 ∞ ∞ 11 11 4 4 3 9 12 3 9
Therefore, permutation P Φ5 acts on the 14 points of set P 1 (Z13 ) the following strange way:
P Φ5
:
1O _
10 o
/
7 2o
/
8_
5
3O _
4o
/
9
11 o
/
12 _ 0e
∞
6e
Chapter 7
75
We have denoted here the points of the finite projective line by their affine coordinates, λ1 (Ak ). We see, that permutation P Φ5 has 2 long orbits, consisting each of 6 points, and has two fixed points. It is useful to observe, that (P Φ5 )6 = 1. Permutation P Φ5 is not a projective transformation. Indeed, otherwise we would have aλ1 , λ5 = cλ1 + d since λ5 = 0 for λ1 = 0. Therefore, condition (λ5 = ∞ for λ1 = 12) would mean that 12c+d = 0, while condition (λ5 = 11 for λ1 = ∞) would mean a = 11c. Thus the projective permutation P Φ5 ought to be λ5 = 11λ1 /(λ1 − 12), which formula would provide for λ1 = 1 the value λ5 = 11/(−11) = 12, contradicting the table at page 74 claiming λ5 (λ1 = 1) = 7. This contradiction shows, that permutation P Φ5 does not preserve the usual projective structure P1 , sending it to a new projective structure P5 , (represented by affine coordinate λ5 on {k (mod 14)}). Permutation P Φ11 permutes the 14 points of set P 1 (Z13 ) (denoted by their coordinates λ1 ) the following (strange) way: P Φ11
:
A/ 8 == == == }
1 ^=
3 _@@
@@ @@ @@
2
/ 10 > == ~ ~ ~ == == ~~~~ } ~
5 ^=
11
/ <12 | | || || }| |
0e
=/ 9 AA }} } AA } AA }}} } ~
4 `AA
∞
6e
7 (two fixed points and four orbits, each consisting of 3 points). Note, that P Φ11 = (P Φ5 )2 and that (P Φ11 )3 = 1. Were the permutation P Φ11 projective, it would have the form λ11 = aλ1 /(cλ1 +d) (since 0 7→ 0), and the action (9 7→ ∞) would imply (9c+d = 0), while the action (∞ 7→ 4) would imply (a = 4c). Thus P Φ11 would have the form 4λ1 . λ11 = λ1 − 9 Therefore, we would obtain the value λ11 (λ1 = 1) = 4/(−8) = −20 = 6 (mod 13) ,
76
Frobenius transformations actions.
contradicting the table’s value λ11 (λ1 = 1) = 8 at page 74. Therefore, nonprojective permutation P Φ11 sends standard projective structure P1 , (associated to affine coordinate λ1 ) to a new projective structure P11 of the same set {k (mod 14)} (associated to affine coordinate λ11 ). Permutation P Φ13 is a projective transformation, since it is generated by the 169 elements field’s Frobenius automorphism Φ13 . Its projectivity is also evident from the λ13 function values in the above table (page 74): noting points of P 1 by their λ1 coordinates, we get the permutation action Case P Φ13 : 1 2 3 4 | | | | 0 6, 5 7 ∞ 12 which diagram implies, that λ13 = −λ1 /(4λ1 + 1). Remaining permutations P Φs (for s = 17 and 23) are now obtainable by the simple multiplications: we note that 13 · 5 = 65 ∈ Γ23 ,
13 · 11 = 143 ∈ Γ17 ,
and therefore hold the identities of permutations products P Φ23 = (P Φ5 )(P Φ13 ) ,
P Φ17 = (P Φ11 )(P Φ13 ) .
Thus, permutation P Φ23 sends structure P1 to structure P5 (the first permutation P Φ13 sends structure P1 to itself, and the next permutation P Φ5 sends P1 transforms it into P5 ). Similarly, permutation P Φ17 sends structure P1 to P11 . Permutation P Φ13 , preserving P1 , preserves also each of the structures P5 and P11 , therefore permutation P Φ23 preserves structure P1 and permutation P Φ17 preserves structure P11 . To study the action of permutation P Φ5 on structure P11 , it suffices to use the affine coordinate λ11 , counting the value of λ11 at point x5 , knowing the value λ11 (x). The permutations product (P Φ11 )(P Φ5 ) being P Φ13 (since 55 ∈ Γ13 ), we get λ11 (x5 ) := λ1 (x55 ) = λ1 (x13 ) := λ13 (x) , and therefore we ought to calculate the dependence of λ13 (x) from λ11 (x). Our table of the values of functions λs (page 74) provide this dependence:
Chapter 7 λ11 (x) λ13 (x)
77 0 0
1 2 3 7 10 9
4 5 6 3 2 6
7 8 8 5
9 10 11 12 ∞ 12 1 4 ∞ 11
Therefore, transformation P Φ5 acts on structure P11 coordinates expression the same way, as it acts on the formula for the structure P1 : the structure P11 is sent to a different projective structure P13 (defined by the affine coordinate λ13 ). But structure P13 coincides with P1 , since Frobenius transformation Φ13 is an automorphism of the 169 elements field. Therefore, the action of transformation Φ5 on the 3 projective structures (generated by the 169 elements field) is provided by the triangular diagram P Φ5
:
P1 aB
BB BB BB B
P11
/
(it sends structure P5 to structure P11 , since 5 · 5 = 25 ∈ Γ11 ). Similarly, one might calculate the action of permutation P Φ11 on these three structures. But one might avoid new calculations, since P Φ11 = (P Φ5 )2 , and, therefore, it acts as the cyclic transformation of order 3, inverse to the action of P Φ5 . We get the triangular diagram of action of P Φ11 on the 3 projective structures, P Φ11
:
P1 `A
AA AA AA
P5
/P < 11 | | | | || }| |
.
The description of the kernel and of the image of projectivization homomorphism ψ : (Γ ≈ {Φs }) −→ {P Φs } , are also implicitly contained in our explicit formulae for permutations P Φs . The answers are the 3 isomorphisms: Γ ≈ (Z6 × Z32 ) ,
78
Frobenius transformations actions. Ker ψ ≈ Z32 ,
Im ψ ≈ Z6 .
Namely, the kernel consists of 8 Frobenius transformations Φs , s being of class Γ1 (where s = 14r + 1). As the kernel’s generators one may take, for instance, transformations Φ29 , Φ43 , Φ85 , taking into account Frobenius transformations relations Φ71 = Φ29 (Φ43 ) ,
Φ113 = Φ29 (Φ85 ) ,
Φ127 = Φ43 (Φ85 ) ,
Φ155 = Φ29 (Φ43 (Φ85 )) . As the generator of the image group one may take the 6-th order permutation g = P Φ5 . The image consists of its powers, namely the following identities hold: g 2 = P Φ11 , g 3 = P Φ13 , g 4 = P Φ37 , g 5 = P Φ17 , g 6 = 1 (since 52 ∈ Γ11 , 53 ∈ Γ13 , 54 ∈ Γ37 , 55 ∈ Γ17 ). Permutation g acts cyclically on the three projective structures P1 , P5 , P11 (generated by the field, consisting of 169 elements). Permutation g 3 leaves every of these 3 structures unchanged, being generated by the field’s Frobenius automorphism. We had thus described all the situation for p = 13. Unfortunately, I have neither theorems, nor even conjectures, extending the above description of the projective geometry to fields with more elements (even in the case of p2 elements with higher primes p, where the geometry remains 1-dimensional). Probably, to find these generalizations one should first calculate the formulae λ = λ(k), describing the projective structures in terms of the projective line coordinate, k (mod (p+1)), and the action of the Frobenius mappings on these structures (at least for the p2 elements fields case) for more examples. The heterogenity of the answers in the cases p = 5, 7, 11 and 13, discussed above, obstructs the attempts to guess the general rules for higher values of p. The author thanks the lecture listeners for many helpful remarks and hopes to continue the collaboration with the readers of the present book, waiting for many contributions to this young domain of mathematics (including, the author hopes, the discovery of extramathematical applications of the Galois fields).
Chapter 8 Cubic tables of fields To facilitate to the readers more experimental studies, I provide here the tables of fields, having 8, 27, 125, 16 or 81 elements. In the case of p3 elements we shall use the additive basis {1, A, A2 }, choosing first some multiplicative group’s generator A. The table fills the cells (u, v, w) of finite cube (tore) Z3p by the degrees k of the powers Ak of multiplicative generator, Ak = uk A2 + v k A + w k 1 . To show this cube’s filling, I shall present below its plane square sections w = const, filled with the degrees k. The presence of the number k (mod (p2 − 1)) in the cell (u, v) of the square w in the table means the identity Ak = uA2 + vA + w1 . These tables are shown below for the square sections of the cube (tore) for p = 2, 3 and 5. Table of the field, consisting of 23 elements. u w=0
1 1 6 0 ∞ 2 0 1
u w=1 v
79
1 5 4 0 0 3 0 1
v
80
Cubic tables of Galois fields The table corresponds to the matrix 0 1 0 (A) = 0 0 1 , 1 0 1
all 6 elements Ak (where 1 ≤ k ≤ 6) are primitive (generators). The table means the 4 identities A3 = A2 + 1, . . . , A6 = A2 + A. These identities are recurrent corollaries of the first one. Table of the field, consisting of 33 elements. v
v
v
2 14 12 6 1 1 19 25 0 ∞ 2 15 0 1 2 u w=0
2 24 20 21 1 18 22 17 0 0 7 16 0 1 2 u w=1
2 5 9 4 1 11 8 23 0 13 3 20 0 1 2 u w=2
The multiplicative generator, that was trix 0 1 (A) = 0 0 2 0
used here, corresponds to the ma 0 1 . 1
The table means 33 − 2 = 25 identities, including, for instance, A0 = 1 , A3 = A2 + 2 , A4 = A2 + 2A + 2 , A5 = 2A + 2 , A6 = 2A2 + 2A , A7 = A2 + 1 , A8 = A2 + A + 2 , A9 = 2A2 + 2A + 2 , . . . , A24 = 2A + 1 , A25 = 2A2 + A , A26 = 1 .
All these identities follow recurrently from the second one of this list (which means, that matrix (A) verifies its own characteristic equation). The number of the multiplicative group’s generators Ak (where 1 ≤ k ≤ 25) equals ϕ(33 − 1) = 12. These 12 values of the degree k are represented in the table by the bold characters.
Chapter 8
81
Table of the field, consisting of 53 elements. u
u
4 63 85 69 112 92
4
22 88 10 35 98
3
32 81 54 61 38
3
6
2
94 100 123 116 19
2 49 58 28 25 14
1
1 30 50
1 29 79 11 71 12
0
∞
2 95 33 34
0
1
2
7 23 3
4
0
v
44 78 83 46
0 55 70 96 51 0
1
w=0
3
4
u 18 118 27 107 121
4 37 109 77 75 114
3 115 4 57 67 103
3
2 122 40 48 105 104
2 53 41 5 119 66
1 99 52 13 15 47
1
80 59 45 89 56
93 65 24 20 39
0
31 101 82 86
3
0
4
0
v
w=1
u 4
2
0
1
2
3
4
v
w=2
60 42 43 110 102
1
2
3
v
w=3
u 4 91 74
9 73 17
0 10 (A) = 0 0 1 3 04
3 111 76 87 90 120 2
68 108 21 16 106
1
84 36 97 72 26
0
62 113 34
8 117
0
3
1
2
4
v
w=4
This table announces 53 − 2 = 123 congruences: A0 = 1 , A3 = 4A2 + 3 , A4 = A2 + 3A + 2 , A5 = 2A2 + 2A + 3 , A6 = 3A + 1 , . . . . . . , A63 = 4A , . . . , A114 = 4A2 + 4A + 3 , . . .
82
Cubic tables of Galois fields A123 = 2A2 + 2A , A124 = 1 .
All these identities are recurrent corollaries of the second one above (which means the characteristic equation verification by matrix (A)). The number of the multiplicative group’s generators Ak (where 1 ≤ k ≤ 123) equals to ϕ(53 − 1) = ϕ(31) ϕ(4) = 60. These 60 values of k are shown in the above table of the field by the bold characters. For the fields, consisting of p4 elements we choose the additive generators 1, A, A2 , A3 , A being a multiplicative group’s generator. The field’s table fills (by degrees k of elements Ak ) cells (u, v, w, t) of finite four-dimensional cube (torus) Z4p , Ak = uk A3 + v k A2 + w k A + t k 1 . To show this filling, we present below its two-dimensional plane sections (w = const, t = const). The appearence of number k (where 1 ≤ k ≤ p4 − 2) at cell (u, v) of the square, numbered (w, t) in the table, means the identity Ak = uA3 + vA2 + wA + t1 . These squares form the following tables, calculated for p = 2 and for p = 3. Table of the field, consisting of 24 elements.
u
u
1 9 11 0 0 1 0 1 v w=0,t=1
1 7 6 0 12 5 0 1 v w=t=1
u
u
1 2 14 0 ∞ 3 0 1 v w=t=0
1 13 8 0 1 10 0 1 v w=1,t=0
Chapter 8
83
This table had been computed for the multiplicative group’s generator, defined in the matricial representation of the field by the matrix 0 1 0 0 0 0 1 0 (A) = 0 0 0 1 . 1 0 0 1 The table above is a geometrical shorthand description of 24 − 3 = 13 identities, A0 = 1 , A 4 = A 3 + 1 , A 5 = A 3 + A + 1 , A6 = A3 + A2 + A + 1 , . . . , A14 = A3 + A2 , A15 = 1 . These identities are recurrent corollaries of the second one above (which is the characteristic equation fulfillment by matrix (A)). The number of the multiplicative group’s generators Ak (where 1 ≤ k ≤ 14) equals to ϕ(24 − 1) = ϕ(3) ϕ(5) = 8. These 8 values of degree k are shown in the table by the bold characters. Table of the field, consisting of 34 elements. v
v
v
2 18 20 36 1 65 27 50 0 40 44 31 0 1 2 u w=0,t=2
2 34 46 7 1 23 73 13 0 68 5 61 0 1 2 u w=1,t=2
2 16 52 15 1 8 62 14 0 37 32 51 0 1 2 u w=2,t=2
v
v
v
2 25 10 67 1 58 70 60 0 0 71 4 0 1 2 u w=0,t=1
2 48 54 22 1 56 55 12 0 77 11 72 0 1 2 u w=1,t=1
2 63 53 33 1 74 47 6 0 28 21 45 0 1 2 u w=2,t=1
v
v
v
2 42 70 39 1 2 79 30 0 ∞ 3 43 0 1 2 u w=0,t=0
2 29 75 64 1 78 57 49 0 1 59 26 0 1 2 u w=1,t=0
2 38 9 17 1 69 24 35 0 41 66 19 0 1 2 u w=2,t=0
84
Cubic tables of Galois fields
This table has been computed for the multiplicative group’s generator, defined (in the matricial representation of the field) by the matrix 0 1 0 0 0 0 1 0 (A) = 0 0 0 1 . 1 0 0 2 The table of the field, containing 81 elements, is the shorthand geometrical version of the 78 = 34 − 3 identities: A0 = 1 , A4 = 2A3 + 1 , A5 = A3 + A + 2 , A6 = 2A3 + A2 + 2A + 1 , A7 = 2A3 + 2A2 + A + 2 , . . . . . . , A79 = A3 + A2 , A80 = 1 . All these identities are recurrent corollaries of the second one in this list (which is the characteristic equation fulfillment by matrix (A)). The number of the multiplicative group’s generators Ak (where 1 ≤ k ≤ 79) equals to ϕ(34 − 1) = ϕ(5) ϕ(16) = 32. These 32 values of the degree k are shown in the table by the bold characters. Tables of the fields, containing 25 , 26 and 27 elements are published in the book: R. Lidl, H. Niederreiter, Finite Fields, Addison–Wesley, 1983 (at pages 673–676 of its second volume). This book contains also a large bibliography of finite fields theory and the proofs of the existence and uniqueness of the field, containing pn elements, as well as of the cyclicity of the multiplicative group of the field, omitted in the present lecture. The tables of the fields, containing 32, 64 and 128 elements are preseneted in this book for the choices of the multiplicative groups’ generators, provided (in the matricial representation of the field) by the following 3 matrices (correspondingly): 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 (A) = 0 0 0 1 0 , , 0 0 0 0 1 0 0 , 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0
Chapter 8
85
implying, corrispondingly, the characteristic equations, encoded in the last line of these 3 Sylvestrian matrices: A5 = A 3 + 1 , A 6 = A 5 + 1 , A 7 = A + 1 . Unfortunately, I do not know convinient form (A) for the multiplicative groups’ generator for a general field, containing pn elements, even for p = 2. The reason of this ignorance might be the difficulty of the bibliographical search, based on the book, quoted above. It mentions, however, that many useful facts appeared first in the book: C. G. J. Jacobi, Canon Arithmeticus, Berlin, 1839, reedited by Academic–Verlag, Berlin, in 1956. However, this pseudofull bibliography attributes important general results of A. Girard (Amsterdam, 1629, “Sur les d´ecouvertes nouvelles en alg`ebre”), used in their book, to I. Newton (1707) and to Waring (neither the name of Girard, nor the bibliographical references to any of these three works being mentioned). The forgotten Girard theorem provides the expression of the moment function (which is the sum of the powers sk = xk1 + · · · + xkn of the roots of the polynomial Y (x − xj ) = xn − σ1 xn−1 + σ2 xn−2 − · · · ± σn ) in terms of the coefficients (predecessing the celebrated Chevalley theorem). The expression is of course an integral coefficients polynomial in the variables σj . These coefficients have many remarkable properties, relating them both to the natural sciences and to number theory (including some generalizations of the “little Fermat theorem” to the matrices traces). The asymptotical behaviour of these coefficients provides the entropy P function pj log pj combinatorial definition (describing the statistics of the long words in a finite alphabet in terms of the frequencies pj of the characters occurences in these words). The same coefficients provide also interesting extensions of the strange “modular”, or “pseudo doubly periodic” p-adic behaviour of the degree d(a, b) of the prime p in the congrueneces pb Cpa − Cab = pd q ,
q being relatively prime to p,
86
Cubic tables of Galois fields
for the binomial coefficients and for their multinomial extension. More details on these congruences and strange periodicities of function d are published in my article “Fermat dynamics of matrices, finite circles and finite Lobachevsky planes”, Cahiers du CEREMADE, Universit´e de ParisDauphine, N◦ 0434 (3 juin 2004) and in the book “Arnold Seminar’s problems, 2004” edited in Russian by the MCCME, Moscow 2005. For instance, d(mp + 1, b) does not depend on b while m is not too large, and d verifies some p-adic periodicity in both arguments a and b. One might guess the nature of function d of two variables a and b, studying the following table of its values for p = 7 (the 14 rows of the Pascal-arranged triangle below correspond to the arguments values 2 ≤ a ≤ 15 for the rows, where 0 < b < a at each row):
5
5
4
3 5
3 4
3 3 5
3 3 4
3 3 3 5
...
3 3 4
4 4 3 3 5
3 5 4 3 4
3 4 4 4 3 5
3 3 5 4 4 4
3 3 4 4 4 4 5
3 3 3 5 4 4 6
3 3 4 4 4 4 5
3 3 5 4 4 4
3 4 4 4 3 5
3 5 4 3 4
4 4 3 3 5
5 3 3 4
3 3 3 5
3 3 4
3 3 5
3 4
3 5
4
5
...
The “double periodicity” of this picture is only a finite repetition of the “fundamental domain” of scale p, which is repeated p times in both directions. Next repetition is disturbed by some corrections. The resulting block of scale p2 is also repeated (p times in each direction), being then disturbed and producing a next block of scale p3 , and so on, but in spite of this padic pseudoperiodicity and of the appearance of pn in the picture, neither the exact p-adic formula for this “periodicity”, nor its relation to the Galois fields (of pn elements) is known. The little Fermat theorem is related to the inequality d ≥ 2 and to some irregular growth of d(a, b) with a. The matricial version of this theorem (and of its generalized form, discovered by Euler, am ≡ am−ϕ(m) (mod m), is the congruence of the traces of integer matrices, tr (Am ) ≡ tr Am−ϕ(m) (mod m)
Chapter 8
87
for m = pn . The relevance of the last condition for this congruence validity suggests its relation to Galois fields (but, as far as I know, such a relation is yet to be discovered). It is a pity, that all these remarkable facts are neglected in modern mathematics and computer science. Numerical experiments helped a lot to discover the relevant empirical facts. Say, the number of the divisors of a large integer n is growing with n, in the average, like its natural logarithm ln n. The sum 2 of the divisors average growth is cn, where c = √ ζ(2) = π /6 ≈ 3/2. The mean divisor average growth is, however, c1 n/( ln n) (rather than cn/ ln n, as a natural scientist would suggest). The last asymptotics had been discovered by A. Karazuba1 , due to my lecturing on the previous Dirichelet results on the averages asymptotics. But no one knows the averages (of the numbers of divisors τ , of their sum σ and of the mean divisor σ/τ ) for the Euler function ϕ values as arguments (that is, for σ(ϕ(n))/τ (ϕ(n)) which would presumably explain the Euler period T (n) averaged asymptotics (T (n) being the minimal period of the geometrical progression of the residues at (t = 1, 2, . . . , T ) modulo n). Empirically this asymptotical growth rate (Cesaro averaged in n) is observed to be c(a)n/(ln n), as had computed F. Aicardi (C.R. Acad. Sci., Paris, ser. I, vol 339 (2004), 15–20: “Empirical estimates of the average order of orbits period lenghts in Euler groups”), considering n . 109 . The minimal period T (n) is a divisor of the Euler function value ϕ(n), and if it might really differ in the average from the Cesaro average growth rate of the mean divisor of ϕ(n) (as the above empirical data suggest). This difference might be explained alternatively either by the fact, that the nature chooses for the period T (n) a nonrandom divisor of ϕ(n), which is far from its mean divisor, or by the fact that the number τ (m) of the divisors of m = ϕ(n), their sum σ(m) and the ratio σ(m)/τ (m) might behave (in Cesaro average) for the arguments m = ϕ(n) very differently, compared to their behaviours for the random arguments m. It might happen, indeed, that the Euler function values ϕ(n) (for ran1
The constant c1 had been then computed by M. Korolev to be c1 ≈ 0, 7138067 . . . : 1 1 Y p3/2 √ ln 1 + c1 = . π p p p−1
88
Cubic tables of Galois fields
dom choices of n) have very different divisors behaviours, (τ (ϕ(n)), σ(ϕ(n)), σ(ϕ(n))/τ (ϕ(n))), comparatively to the averaged divisors behaviours for the random numbers n themselves (τ (n), σ(n), σ(n)/τ (n)). These (alternative?) possibilities of the nonrandomness of the statistics of the average divisors of ϕ(n) or of the special divisor T (n) of ϕ(n) are both possible. Whether they really occure, deserve both the empirical study and mathematical theories. But these subjects seem to be too classical to attract modern mathematicians. The Girard name and theory are not mentioned even in the celebrated “Concrete Mathematics” textbook, whose authors wished to unify the continuous mathematics (whence the ”con...” part of “concrete”) with the discrete one (whence the ”...crete” part). Considering the unity of mathematics to be its main jewel, I hoped to contribute (by the geometrical presentation of the Galois fields and of their relations to ergodic theory of dynamical systems, to statistics and to projective geometry) to the return of all these forgotten classical theories to the continuous real (R) world of the natural science with the present lecture.
