Polynomial characterization of a finite field
= Polynomial characterization of a finite field
{title2=$X^{q^n}-X$}
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 $\mathbb F_q$.