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\title{Math 721\\Spring 2011\\Homework 6}
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\centerline{\it Due Friday May 6, 2011.}

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\noindent 1. (5 pt) Let $F$ be a finite field. Show that any element in $F$ can be written as the sum of two squares.

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\noindent 2. Let $K$ be a finite field and $F$ an algebraic closure of $K$. 
\begin{itemize}
\item[a)] (5 pt) Show that $\text{Gal}(F/K)$ is abelian.
\item[b)] (5 pt) Show that every nonidentity element of $\text{Gal}(F/K)$ is of infinite order.
\end{itemize}

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\noindent 3. Let the characteristic of $K$ be $p>0$. 
\begin{itemize}
\item[a)] (5 pt) If $[F:K]=n$ and $p$ does not divide $n$, then $F$ is separable over $K$.
\item[b)] (5 pt) If $u\in F$ is algebraic over $K$, then $u$ is separable over $K$ if and only if $K(u)=K(u^{p^n})$ for all $n\geq 1$.
\end{itemize}

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\noindent 4. (5 pt) Let $F$ be an extension of $K$ and $u,v\in F$ such that $u$ is separable over $K$ and $v$ is totally inseperable over $K$. Show that $K(u+v)=K(u,v)$. Also show that $K(u,v)=K(uv)$ if both $u$ and $v$ are nonzero.





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