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**Additional info for A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics, Volume 84)**

**Example text**

2 (Abelian Group). An abelian group is a group G such that ab = ba for every a, b ∈ G. 3 (Ring). A ring R is a set equipped with binary operations + and × and elements 0, 1 ∈ R such that R is an abelian group under +, and for all a, b, c ∈ R we have • 1a = a1 = a • (ab)c = a(bc) • a(b + c) = ab + ac. If, in addition, ab = ba for all a, b ∈ R, then we call R a commutative ring. In this section, we define the ring Z/nZ of integers modulo n, introduce the Euler ϕ-function, and relate it to the multiplicative order of certain elements of Z/nZ.

The ϕ(ϕ(9)) = ϕ(6) = 2 primitive roots modulo 9 are 2 and 5. There are no primitive roots modulo 8, even though ϕ(ϕ(8)) = ϕ(4) = 2 > 0. 14 (Emil Artin). Suppose a ∈ Z is not −1 or a perfect square. Then there are infinitely many primes p such that a is a primitive root modulo p. 44 2. The Ring of Integers Modulo n There is no single integer a such that Artin’s conjecture is known to be true. For any given a, Pieter [Mor93] proved that there are infinitely many p such that the order of a is divisible by the largest prime factor of p − 1.

Suppose a ∈ Z is not −1 or a perfect square. Then there are infinitely many primes p such that a is a primitive root modulo p. 44 2. The Ring of Integers Modulo n There is no single integer a such that Artin’s conjecture is known to be true. For any given a, Pieter [Mor93] proved that there are infinitely many p such that the order of a is divisible by the largest prime factor of p − 1. 14. 15. Artin conjectured more precisely that if N (x, a) is the number of primes p ≤ x such that a is a primitive root modulo p, then N (x, a) is asymptotic to C(a)π(x), where C(a) is a positive constant that depends only on a and π(x) is the number of primes up to x.