Math 6310 • Algebra

Let \(f(x)\) be a degree 5 polynomial in \(\mathbb{Q}[x]\) that is not solvable by radicals. Let \(L\)
be its splitting field over \(\mathbb{Q}\).
\begin{enumerate}
\item[(i)]
Prove that there is at most one field \(K\) with \(\mathbb{Q}\subset K\subset L\) and
\([K:\mathbb{Q}]=2\).

\item[(ii)]
If \(\alpha,\beta\) are irrational elements in \(L\) such that \(\alpha^2\) and \(\beta^2\) are rational,
prove that \(\alpha\beta\) is rational.
\end{enumerate}