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\title{Math 81/111: Rings and Fields \\ Homework \#8}
\date{25 February 2015; due Monday, 9 March 2015}

\maketitle

This homework set is required for students enrolled in Math 111 and is optional (strongly encouraged) for those in Math 81.

\vspace{2ex}

\begin{problab}{8.1}
Let $\Omega/F$ be an extension of fields and let $G=\Aut_F(\Omega)$ be equipped with the Krull topology, with the basis of open sets
\[ V(\sigma,S) = \{\tau \in G: \tau|_S=\sigma|_S\} \]
for $\sigma \in G$ and $S \subseteq \Omega$ finite.
\begin{enumalph}
\item Show that the inverse map $G \to G$ by $\sigma \mapsto \sigma^{-1}$ is continuous.
\item Let $X \subseteq G$ be a subset.  Show that the closure $\cl(X)$ of $X$ in $G$ is equal to
\[ \cl(X)=\{\sigma \in G : \text{for all finite $S \subseteq \Omega$, there exists $\tau \in X$ such that $\tau|_S=\sigma|_S$}\}. \]
\item Let $H \leq G$ be a subgroup (not necessarily closed).  Show that $H$ and $\cl(H)$ have the same fixed subfield, i.e.,
\[ \Omega^H=\Omega^{\cl(H)} \subseteq \Omega. \]
\end{enumalph}
\end{problab}

\begin{problab}{8.2}
Consider the direct product of groups
\[ \prod_{n=1}^{\infty} \Z/n\Z \]
equipped with the product topology and each $\Z/n\Z$ the discrete topology.  Let
\[ \widehat{\Z}=\{(a_n)_{n \geq 1} : a_n \equiv a_m \psmod{m} \text{ if $m \mid n$}\} \subseteq \prod_{n=1}^{\infty} \Z/n\Z. \]
\begin{enumalph}
\item Show that $\widehat{\Z}$ (with the subspace topology) is a compact subgroup of $\prod_{n=1}^{\infty} \Z/n\Z$.
\item Show that the map $\Z \to \widehat{\Z}$ by $1 \mapsto (1,1,\dots)$ is a group homomorphism with dense image, giving $\Z$ the subspace topology.
\item Using the map
\[ \Gal(\overline{\F_q}/\F_q) \hookrightarrow \prod_{n=1}^{\infty} \Gal(\F_{q^n}/\F_q) \] 
show that there is an isomorphism of topological groups
\[ \Gal(\overline{\F_q}/\F_q) \xrightarrow{\sim} \widehat{\Z} \]
i.e., a group homomorphism that is also a homeomorphism of topological spaces.  \emph{[Hint: See also Example 7.16 for some inspiration.]}
\end{enumalph}
\end{problab}

\begin{problab}{8.3}
Let $\Omega/F$ be a Galois extension of fields (finite or infinite).  Show that $\Gal(L/K)$ is either finite or uncountable.
\end{problab}

\end{document}