\documentclass{amsart}
\title{Algebra 2. Problem Set 10}
\author{}

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\begin{document}
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\problem{1} Let $E/F$ be an algebraic extension. Let $\sigma: E\to E$ be a morphism
of fields such that $\left. \sigma \right|_{F} = \Id_F$. Prove that $\sigma$
is then an isomorphism.\\

\problem{2} Give an example to prove that the assertion of Problem 1 is false, if
$E/F$ is not assumed to be algebraic.\\

\problem{3} Let $E/F$ be a normal extension. Let $K$ be an intermediate field: 
$F\subset K\subset E$ and let $\sigma: K\to E$ be an embedding which is the identity
on $F$. Prove that $\sigma$ extends to an automorphism of $E$.\\

\problem{4} Let $F = \Q$, $E$ be the splitting field of $X^2-2\in \Q[X]$ over $\Q$.
Thus $E = \Q(\sqrt{2})$.
Let $K$ be the splitting field of $X^2-\sqrt{2}\in E[X]$ over $E$. Describe the
group $\gal{K}{F}$. Prove that $K/F$ is not a Galois extension.\\

\problem{5} Let $E = \Q(\sqrt{2},\sqrt{3})$ and $F = \Q$. Compute $\gal{E}{F}$.
Is $E/F$ a Galois extension?\\

\problem{6} Let $\alpha = \sqrt{2+\sqrt{2}} \in \R$, and let $E = \Q(\alpha)$
(with $F = \Q$). Compute $\gal{E}{F}$.\\

\problem{7} For a finite extension $E/F$, prove that $\gal{E}{F}$ is finite.\\

\problem{8} Let $\zeta = \exp\lp \frac{2\pi\iota}{7}\rp \in \C$. Prove that
$\Q(\zeta)$ is a Galois extension of $\Q$.\\

\problem{9} Let $k\subset K$ be two fields, and assume that $K$ is algebraically closed.
Let $K'$ consist of all elements of $K$ which are algebraic over $k$. Prove
that $K'$ is isomorphic to the algebraic closure of $k$.\\

\problem{10} Consider the field $E = \Q(T_1,\ldots, T_n)$. For each $k\in \{1,\ldots,n\}$
define:
\[
\varepsilon_k := \sum_{1\leq i_1<i_2<\cdots < i_k\leq n} T_{i_1}T_{i_2}\cdots T_{i_k}
\]
and let $F = \Q(\varepsilon_1,\ldots,\varepsilon_n)\subset E$. Prove that $(E:F) = n!$
in the following steps.

\begin{enumerate}
\item Prove that $E/F$ is the splitting extension of a degree $n$ polynomial. ({\em
Hence by Problem 3 of Problem Set 9, $(E:F)$ divides $n!$.})\\

\item Let $S_n$ be the permutation group on $n$ letters, acting on $E$ in the
obvious way. Prove that $F\subset E^{S_n}$. ({\em Hence, $(E:F) \geq (E:E^{S_n}) = n!$
by Artin's theorem.})
\end{enumerate}

\end{document}
