Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 18I / a / i

An irreducible degree-$n$ polynomial over ${\mathbb{F}}_q$ is separable and its roots lie in ${\mathbb{F}}_{q^n}$, with any one root generating that field. Hence its splitting field is ${\mathbb{F}}_{q^n}$ andgenerated by the Frobenius automorphism $x\mapsto x^q$.