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
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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.
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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.
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