Math 6310 • Algebra

Let \(E\) be an extension of the field \(F\) and let \(\alpha\) be an element of \(E\) which is algebraic
over \(F\) of degree \(n\).
\begin{enumerate}
\item[(a)] If \(n\) is prime to 2, prove that \(F(\alpha^2)=F(\alpha)\).
\item[(b)] Give an example in which \(n\) is prime to 3 but \(F(\alpha^3)\ne F(\alpha)\).
\end{enumerate}