Math 6310 • Algebra

Let \(p\) be a prime number and \(\mathbb{F}_p\) be the field with \(p\) elements.
Let \(f(x)=x^p-x-1\in\mathbb{F}_p[x]\) and let \(K\) be the splitting field of \(f(x)\). Show that:
\begin{enumerate}
\item[(a)] if \(\alpha\) is a root of \(f(x)\), then \(\alpha+1\) is a root of \(f(x)\),
\item[(b)] \(K=\mathbb{F}_p(\alpha)\),
\item[(c)] the Frobenius automorphism \(\sigma(x)=x^p\) is a non-trivial element of
\(\operatorname{Gal}(K/\mathbb{F}_p)\),
\item[(d)] \(f(x)\in\mathbb{F}_p[x]\) is irreducible.
\end{enumerate}