Math 6310 • Algebra

Let \(f(x)\) be a polynomial of degree \(n>2\) over \(\mathbb{Q}\), the field of rational numbers.
Let \(K\) be the splitting field of \(f(x)\) over \(\mathbb{Q}\) and suppose that the Galois group of
\(f(x)\) over \(\mathbb{Q}\) is \(S_n\), the group of all permutations of \(n\) objects. Prove the
following:
\begin{enumerate}
\item[(a)] \(f(x)\) is irreducible over \(\mathbb{Q}\).
\item[(b)] Let \(\alpha\) be a root of \(f(x)\) in \(K\) and let \(\sigma\) be an automorphism of the
field \(\mathbb{Q}(\alpha)\). Prove that \(\sigma=1\).
\end{enumerate}