Let $\mathbf{F}_{p}$ be the finite field with $p$ elements ( $p$ a prime), and let $k$ be a finite extension of $\mathbf{F}_{p}$. Define the Frobenius automorphism $\sigma: k \longrightarrow k$, verifying that it is an $\mathbf{F}_{p^{-}}$ automorphism of $k$.

Suppose $f=X^{p+1}+X^{p}+1 \in \mathbf{F}_{p}[X]$ and that $K$ is its splitting field over $\mathbf{F}_{p}$. Why are the zeros of $f$ distinct? If $\alpha$ is any zero of $f$ in $K$, show that $\sigma(\alpha)=-\frac{1}{\alpha+1}$. Prove that $f$ has at most two zeros in $\mathbf{F}_{p}$ and that $\sigma^{3}=i d$. Deduce that the Galois group of $f$ over $\mathbf{F}_{p}$ is a cyclic group of order three.