Math 6310 • Algebra

\begin{enumerate}
\item[(a)]
Suppose \(K\) is a field of prime order and \(E\) is the algebraic closure of \(K\). If \(F\) is any
subfield of \(E\), prove that the multiplicative group of \(F\) is periodic, i.e., every element of
the multiplicative group has finite order.
\item[(b)]
Suppose \(F\) is a field whose multiplicative group is periodic. Prove that for some prime \(p\),
\(F\) is isomorphic to a subfield of the algebraic closure of the field of order \(p\).
\end{enumerate}