Proposition 1

— Let K be a finite field with q elements.

a) The characteristic of K is a prime number p, and there exists an integer f \geq 1 such that q = p^f.

b) The additive group of K is a direct sum of f cyclic groups of order p.

c) The multiplicative group of K is cyclic of order q - 1.

Since Z is infinite and K finite, the unique ring homomorphism \varphi : Z \to K is not injective and its kernel is a prime ideal of Z not equal to 0. Therefore the characteristic of K is a prime number and K is an algebra over the field \mathbf{F}_p of p elements (V, p. 2). Let f be the degree of K over \mathbf{F}_p. If f were infinite, K would contain, for each integer n \geq 0 a subspace of dimension n over \mathbf{F}_p, whence q \geq p^n, which is absurd. Therefore f is finite. The additive group of K is thus isomorphic to (\mathbf{F}_p)^f, whence the assertions a) and b).

Assertion c) follows by Lemma 1 (V, p. 78).