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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 11.2  \\ Fall 2019 \hfill Due December 2}}}

\large

\vspace{1cm}

\begin{exercise}
Let $K/k/F$ be a tower of fields.  If $\alpha \in K$ is algebraic over $F$, show that
\[
[k(\alpha) : k] \le [F(\alpha) : F].
\]
\end{exercise}

\begin{exercise}
Let $K/F$ be an extension of fields and suppose $\alpha_1, \ldots, \alpha_n \in K$ are algebraic over $F$.  Prove that
\[
[F(\alpha_1, \ldots, \alpha_n) : F] \le \prod_{i=1}^n [F(\alpha_i):F].
\]
In particular, $F(\alpha_1, \ldots, \alpha_n)$ is a finite extension of $F$.  [{\em Suggestion:}  Induct on $n$.]
\end{exercise}

\begin{exercise}
Let $K/F$ be fields and let $\alpha \in K$.  Prove that if $\alpha$ is algebraic of odd degree over $F$, then so is $\alpha^2$ and $F(\alpha) = F(\alpha^2)$.
\end{exercise}


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