CHAPTER 1 Arithmetic and the symmetric group his chapter’s aim is to recall some results from arithmetic and to provide T some of their applications in computer science. We shall also give a couple of reminders about the symmetric group that we constantly use in Galois theory. 1.1 Reminder of arithmetic 1.1.1 Ring Z / nZ We recall here some results from arithmetic that are often used in the following chapters. First, let us quote Bézout’s identity, an easy but fundamental lemma: if a and b are two coprime integers, then there exist two integers u and v such that au + bv = 1. Given an integer n N×, we recall that the set of congruence classes modulo n is the set 0, 1,..., n∈ 1 , denoted by Z / nZ. This set is naturally endowed with a commutative{ ring− structure.} The generators of Z / nZ (as an additive group) are the classes m such that m and n are coprime. The additive group Z / nZ is isomorphic to the multiplicative group of nth complex roots of unity. Let us also recall that a class m is invertible if and only if m and n are coprime, and that, otherwise, m divides 0. Finally, Z / pZ is a field if and only if p is a prime number; in that case, we set Fp = Z / pZ. 810151_.pdf 21 10/14/2016 10:52:42 AM 4 CHAPTER 1. ARITHMETIC AND THE SYMMETRIC GROUP The Chinese remainder theorem asserts that if m and n are coprime, then the rings Z / mnZ and Z / mZ Z / nZ are isomorphic. × 1.1.2 Groups and Euler’s totient function Let G be a group whose identity element is denoted by e. We recall that an element g G has finite order if there exists an integer n N× such that n ∈ n ∈ g = e. The smallest integer n N× such that g = e is called the order of g and denoted by o(g). The cardinality∈ of a finite group is also called the order of the group. This notion is the same as the previous definition of the order in the case of a finite cyclic group, i.e., a finite group that is generated by a single element. The basic (but fundamental) result about the order is Lagrange’s theorem, which states that the order of every element in a finite group divides the group’s order. Let us provide an application of this theorem that will be used later: if two elements g1 and g2 commute and have finite and coprime orders m1 and m2, then the order of g1g2 is the product of their orders. In fact, the m1m2 identity (g1g2) = e shows that the order of g1g2 divides m1m2. Con- q q q versely, if q N× satisfies (g g ) = e, the element g− = g belongs to the ∈ 1 2 2 1 group g1 g2 . By Lagrange’s theorem, its order divides m1 and m2, thus it h i ∩ h i q is equal to 1, and we have g1 = e. Therefore, m1 divides q; in the same way, m2 divides q, thus m1m2 divides q. Let us also quote another interesting result that we will use later: Cauchy’s theorem asserts that if a prime number p divides the order of a finite group G, then G contains an element of order p. Let now G be a finite group that acts on a finite set X. For every x X, we denote by G x the orbit of x and G the stabilizer of x in G. The map ψ∈from G · x to G x defined by ψ(g) = g x induces a one-to-one correspondence from G/Gx · · G onto G x, hence the identity G x = |G | . The set of all orbits is a partition of · | · | | x| X, thus if we denote by O1,...,Or the distinct orbits of X under the action of G, then we have X = O + + O . This formula is called the class equation. | | | 1| ··· | r| For every g G, we define the stabilizer of g by fixg := x X / g x = x . Then we have∈ Burnside’s lemma: { ∈ · } 1 r = fix . G | g| g G | | X∈ 810151_.pdf 22 10/14/2016 10:52:42 AM 1.1. REMINDER OF ARITHMETIC 5 In fact, let us consider the set F := (g, x) G X / g x = x . Then we { ∈ × · } have F = fix , and | | | g| g G X∈ r r G F = Gx = Gx = | | = r G , | | | | | | Oj | | x X j=0 x O j=0 x O X∈ X X∈ j X X∈ j | | and we get the result. For every n N 0, 1 , we denote by ϕ(n) the number of invertible elements in Z / nZ. We∈ also\{ set}ϕ(1) = 1. The Euler’s totient function is the map ϕ from N× to N× such that the number ϕ(n) is the order of the group (Z / nZ)×. It follows from the Chinese remainder theorem that if n1, . , nr are numbers that are pairwise coprime, then ϕ(n1 . nr) = ϕ(n1) . ϕ(nr). If p is a prime number, then Z / pZ is a field, thus ϕ(p) = p 1. If α N, then ϕ(pα) is the number of elements in the set [[0, pα 1]] that are− coprime∈ to pα. These elements α − α 2 α are the elements of the set [[0, p 1]] 0, p, 2p, . , p − p , hence ϕ(p ) = α α 1 − \{ α1 } α p p − . Finally, if n N× can be written as n = p . p r , where the − ∈ 1 r p1, . , pr are distinct prime numbers, then r r α α 1 1 ϕ(n) = p j p j − = n 1 . j − j − p j=1 j=1 j Y Y Let a be an integer that is coprime to n. Then a belongs to (Z / nZ)×, thus by Lagrange’s theorem, aϕ(n) = 1, i.e., aϕ(n) 1 mod n. This result is called Euler’s theorem. Fermat’s little theorem follows≡ if we take a prime number p: p 1 in this case, we have ϕ(p) = p 1, thus if a and p are coprime, then a − 1 mod p. − ≡ Finally, let us provide an application of Fermat’s little theorem that will be useful later. Lemma 1.1.1 Let a, l be two integers and p, q be two distinct prime numbers. 1+l(p 1)(q 1) Then a − − a mod pq. ≡ Proof: First, let us consider two cases. l(p 1)(q 1) l(q 1) p 1 — If p does not divide a, then a − − = a − − 1 mod p according to Fermat’s little theorem. ≡ 1+l(p 1)(q 1) — If p divides a, then a − − 0 a mod p. ≡ ≡ 810151_.pdf 23 10/14/2016 10:52:42 AM 6 CHAPTER 1. ARITHMETIC AND THE SYMMETRIC GROUP 1+l(p 1)(q 1) Therefore, in both cases we have a − − a mod p. In the same way, 1+l(p 1)(q 1) ≡ we have the relation a − − a mod q. This shows that both p and q 1+l(p 1)(q 1) ≡ divide a − − a. As p and q are distinct prime numbers, their prod- − 1+l(p 1)(q 1) uct pq also divides the number a − − a and we are done. − We have the relation n = ϕ(d), called Euler’s formula. In fact, for every d n 1 X| positive divisor d of n, let us set Fd := k [[0, n 1]] / k n = d n n { ∈ − k ∧ } and F ′ := k [[0, 1]] / k = 1 . Then the map k is a one-to-one d { ∈ d − ∧ d } 7→ d correspondence from Fd onto Fd′ , whose inverse function is the map l dl. n 7→ Therefore, Fd = Fd′ = ϕ d . As the set Fd / d n is a partition of [[1, n]], we have | | | | { | } n ϕ(e) = ϕ = F = n. d | d| e n d n d n X| X| X| Let us recall a result about the structure of finite cyclic groups. Its proof makes use of the previously stated methods. Proposition 1.1.2 If G is a finite cyclic group of order n, then for every divisor d of n, G possesses a unique subgroup of order d. Proof: Let G := x be a finite cyclic group of order n. For every j [[0, n 1]], the hj i n ∈ − order of x is n j . Then the number of elements of order d is the cardinality ∧ n d of the set j [[0, n 1]] / j n = d . Moreover, the map j j n is a one- { ∈ − ∧ } n 7→ to-one correspondence (its inverse function is l l d ) between this set and the set l [[0, d 1]] / l d = 1 . Hence the number7→ of elements of order d is ϕ(d). n { ∈ − d∧ } n Furthermore, y := x has order n n = d. Let H := y be the subgroup of G d h i generated by y. Then the number∧ of elements of order d in H is the cardinality of the set l [[0, d 1]] / l d = 1 , i.e., ϕ(d). Thus H contains all the elements of order d{. ∈ − ∧ } The following theorem is a consequence of Euler’s formula. Theorem 1.1.3 Every finite subgroup of the multiplicative group of a field K is cyclic. n In particular, for n N×, the set of roots of X 1 in K is a finite cyclic ∈ − subgroup of K× whose order divides n.