Proposition 2

— Let K be a finite field of q elements. The field K is a splitting field of the polynomial X^q - X of \mathbf{F}_p[X] and it is the set of all roots of this polynomial.

1 In agreement with the conventions of this chapter we shall only be concerned here with finite commutative fields. In fact, every finite field is commutative, as we shall see in Chapter VIII (cf. V, p. 170, Exercise 14).

For every $x \neq 0$ in $K$ we have $x^{q-1} = 1$ because $K^*$ is a finite group of order $q-1$ (I, p. 52). It follows that $x^q = x$ for each $x$ in $K$. The polynomial $X^q - X$ of $\mathbf{F}_p[X]$ is of degree $q$ and it has $q$ roots in $K$, whence

Proposition 2 follows at once from this.