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\begin{center}{\bf M901, Assignment 7: Due Monday, November 21, 2011}\end{center}

{\it Instructions}: Do any three problems.

Background: A field $k$ is said to be {\it perfect} if either $\operatorname{char}(k)=0$, or
$\operatorname{char}(k)=p>0$ and every element of $k$ is the $p$th power 
of some element of $k$ (i.e., $k^p=k$, where we define $k^p=\{a^p:a\in k\}$).


\begin{itemize}

\item[(1)] Let $k$ be a field. If $k$ is either finite or algebraically closed, show that $k$ is perfect.

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\item[(2)] Let $k$ be a field with $\operatorname{char}(k)=p>0$. Show that the subset
$k^{\frac{1}{p}}=\{a\in k^a: a^p\in k\}$ is a subfield of $k^a$ isomorphic to $k$ but containing $k$.
Give an example where $k\subsetneq k^{\frac{1}{p}}$. (Note that $(k^{\frac{1}{p}})^p=k$,
so your example will give $(k^{\frac{1}{p}})^p=k\subsetneq k^{\frac{1}{p}}$;
i.e., $k^{\frac{1}{p}}$ will be an example of a non-perfect field.)

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\item[(3)] Let $k$ be a field, $\operatorname{char}(k)=p>0$. Show that the following are equivalent:
\begin{itemize}
\item[(a)] $k$ is a perfect field;
\item[(b)] every irreducible polynomial $f(x)\in k[x]$ is separable; and
\item[(c)] every algebraic extension of $k$ is separable.
\end{itemize}

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\item[(4)] Let $k$ be a field with $\operatorname{char}(k)=p>0$ and let $a$ be algebraic over $k$.
Prove that $a$ is separable over $k$ if and only if $k(a)=k(a^{p^n})$ for every integer $n\geq 1$.

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\item[(5)] Let $k$ be a field with $a$ and $b$ algebraic over $k$ and $a$ separable over $k$.
Prove that $k(a,b)$ is simple over $k$.

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\item[(6)] Let $k$ be a field with $\operatorname{char}(k)=p>0$. Let $x$ and $y$ be indeterminates.
Prove that $k(x^p,y^p)\subset k(x,y)$ has infinitely many distinct intermediate fields.

\end{itemize}
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