Introduction to Number Theory Luigi Schiavone based on lessons of Ivan Visconti, full professor at Università degli Studi di Salerno, Italy A.A. 2021/2022 Contents 1 Basic Group Theory 1.1 Modulo operator . . . . . . . 1.2 Groups . . . . . . . . . . . . . 1.3 The group (ZN , +) . . . . . . 1.4 The group (ZP∗ , ∗) . . . . . . . 1.5 Properties of finite groups and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cyclic groups 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 2 3 3 3 1 Basic Group Theory This is a review of prime numbers and basic modular arithmetic. 1.1 Modulo operator Definition Let a, N ∈ Z with N > 1. We use the notation [a mod N ] to denote the remainder of a upon division by N . Modulo equality Let a, b, N ∈ Z with N > 1. We use the notation [a = b mod N ] to denote that remainder of a upon division by N is equal to the remainder of b upon division by N . 1.2 Groups Let G be a set. A binary operation ◦ on G is simply a function ◦(·, ·) that takes as input two elements of G. If g, h ∈ G then instead of using the cumbersome notation ◦(g, h), we write g ◦ h. Definition A group (G, ◦) is a set of elements G and a binary operation ◦ with the following properties: • Closure: For all g, h ∈ G, g ◦ h ∈ G. • Existence of an identity: There exists an identity e ∈ G such that for all g ∈ G, e ◦ g = g =g◦e • Existence of inverses: For all g ∈ G there exists an element h ∈ G such that g ◦ h = e = h ◦ g. Such an h is called an inverse of g. • Associativity: For all g1 , g2 , g3 ∈ G, (g1 ◦ g2 ) ◦ g3 = g1 ◦ (g2 ◦ g3 ). Definition A group (G, ◦) is abelian if the following property holds: • Commutativity: For all g, h ∈ G, g ◦ h = h ◦ g Definition A group (G, ◦) is finite if G has a finite number of elements. We call |G| the order of the group. We will always deal with finite, abelian groups. Definition If (G, ◦) is a group, H ⊆ G and (H, ◦) is a group then we call (H, ◦) a subgroup of G. Example 1. ({e}, ◦) is a subgroup of (G, ◦). It’s also abelian and finite. 2 1.3 The group (ZN , +) Definition Let N > 1 be an integer. We define the group (ZN = {0, ..., N − 1}, +) where +(a, b) = a + b mod N . Theorem. (ZN , +) is an abelian finite group of order N . Proof. Closure is obvious; associativity and commutativity follow from the fact that the integers satisfy these properties; the identity is 0; and, since a + (N − a) mod N = 0, it follows that the inverse of any element a is (N − a). With the notation ka mod N with k ∈ Z we are denoting ( Pk 1.4 i=1 a) mod N ; 0a mod N = 0. The group (ZP∗ , ∗) Definition Let P > 1 be a prime integer. We define the group (ZP∗ = {1, ..., P − 1}, ∗) where ∗(a, b) = a ∗ b mod P . Theorem. (ZP∗ , ∗) is an abelian finite group of order P − 1. Proof. Closure is obvious; associativity and commutativity follow from the fact that the integers satisfy these properties; the identity is 1 since a ∗ 1 mod P = a mod P = a; we’ll see that also the inverse exists. With the notation ak mod P with k ∈ Z we are denoting ( we denote with a−1 the inverse of a. Qk i=1 a) mod P ; a0 mod N = 1; The following holds: ac bc mod P = (ab)c mod P and ac ad mod P = ac+d mod P , where we omitted the symbol ∗. 1.5 Properties of finite groups and cyclic groups The following holds: if (G, ◦) is a finite group of order m then ∀a ∈ G doing the operation ◦ m − 1 times involving a m times results in the identity element e. Therefore ∀a ∈ G doing the operation ◦ x − 1 times results the same of doing it x − 1 mod m times. Example 2. Consider (ZP∗ , ∗). Then ∀a ∈ ZP∗ aP −1 = 1 mod P and also ax = ax mod P −1 mod P Example 3. Consider(ZN , +). Then ∀a ∈ ZN aN = 0 mod N and also ax = a(x mod N ) mod N Definition A cyclic group (G, ◦) is a finite group of order m such that ∃g ∈ G : < g >= G where < g > is the set of elements obtained by iterating the group operator over g. g is called a generator of G. The following holds: if (G, ◦) is a finite group of order m then | < a ∈ G > | is a divisor of m; if (G, ◦) is a finite group of order m with m prime then it is cyclic and each element of G excluding i is a generator. Note that if p = 2q + 1 with q, p prime, then a cyclic subgroup of ZP∗ of prime order larger ≈ the half of ZP∗ exists. 3