Let $F$ be a finite field. Show that there is a unique prime $p$ for which $F$ contains the field $\mathbb{F}_{p}$ of $p$ elements. Prove that $F$ contains $p^{n}$ elements, for some $n \in \mathbb{N}$. Show that $x^{p^{n}}=x$ for all $x \in F$, and hence find a polynomial $f \in \mathbb{F}_{p}[X]$ such that $F$ is the splitting field of $f$. Show that, up to isomorphism, $F$ is the unique field $\mathbb{F}_{p^{n}}$ of size $p^{n}$.

[Standard results about splitting fields may be assumed.]

Prove that the mapping sending $x$ to $x^{p}$ is an automorphism of $\mathbb{F}_{p^{n}}$. Deduce that the Galois group Gal $\left(\mathbb{F}_{p^{n}} / \mathbb{F}_{p}\right)$ is cyclic of order $n$. For which $m$ is $\mathbb{F}_{p^{m}}$ a subfield of $\mathbb{F}_{p^{n}}$ ?