Proof

Since p is a prime, it follows from Section 2.1 that (Z/pZ)^* is a (multiplicative) group of order p - 1. Hence the order of every element x is a divisor of p - 1, so that x^{p - 1} = 1. This just says that for every n which is not divisible by p the relation n^p = 1 mod p holds.