Let F be a field. The following lemma is concerned with polynomials in F[X].

Lemma

Suppose that m and n are positive integers. Then

gcd(Xm - 1,Xn - 1) = Xgcd(m,n) - 1.

In particular, if m | n, then (Xm - 1) | (Xn - 1).

We use this observation to prove the following result, announced before.

Theorem

The multiplicative group of a finite field of order q is cyclic of order q - 1.

In terminology introduced before, the theorem says that any field has a primitive element.