The Structure of U(ℤ/nℤ)

Abstract

Having introduced the notion of congruence and discussed some of its properties and applications we shall now go more deeply into the subject. The key result is the existence of primitive roots modulo a prime. This theorem was used by mathematicians before Gauss but he was the first to give a proof. In the terminology introduced in Chapter 3 the existence of primitive roots is equivalent to the fact that U(ℤ/pℤ) is a cyclic group when p is a prime. Using this fact we shall find an explicit description of the group U(ℤ/nℤ) for arbitrary n.