 Polynomial characterization of a finite field

In a splitting field over ${\mathbb{F}}_q$, the polynomial $X^{q^n}-X$ has $q^n$ distinct roots because its derivative is $-1$. Its roots are closed under addition, multiplication, additive inverses, and nonzero inverses, so they form the field ${\mathbb{F}}_{q^n}$. The splitting field consequently has degree $n$ over ${\mathbb{F}}_q$.