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\parbox{1.95in}{\bf Math 4720/5210}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{1.95in}{\bf \hfill Due: Wed., Apr. 3${}^\mathrm{\bf rd}$, 2024}

\vspace{0.2in}

\begin{description}
\item[\#1 The Good and the Bad:] Consider the following extensions of $\mathbb{Q}$: $\mathbb{E}_1 = \mathbb{Q}[\sqrt{9}]$, $\mathbb{E}_2=\mathbb{Q}[\sqrt{8}]$, and $\mathbb{E}_3=\mathbb{Q}[\sqrt[3]{7}]$. 
\begin{enumerate}[(a)]
\item Compute and carefully justify these extensions' degrees: $[\mathbb{E}_1:\mathbb{Q}]$, $[\mathbb{E}_2:\mathbb{Q}]$, and $[\mathbb{E}_3:\mathbb{Q}]$.
\item Explicitly write down the automorphisms (including actual formulas) making up each of the Galois groups: $G_1=\mathrm{Gal}(\mathbb{E}_1/\mathbb{Q})$, $G_2=\mathrm{Gal}(\mathbb{E}_2/\mathbb{Q})$, and $G_3=\mathrm{Gal}(\mathbb{E}_3/\mathbb{Q})$. 

Carefully explain why $G_1$ and $G_3$ are trivial groups whereas $G_2$ is not. 
\item Suppose we modify the second extension so we have $\mathbb{E}_{2'} = \mathbb{F}[\sqrt{8}]$ extending $\mathbb{F}$ for some subfield of $\mathbb{C}$. Is it possible that $G_{2'}=\mathrm{Gal}(\mathbb{E}_{2'}/\mathbb{F})$ has a different group structure? Can $G_{2'}$ be a bigger group? A smaller group?
\end{enumerate}

\item[\#2 Doing My Job:] Let $\mathbb{E}/\mathbb{F}$ be an extension of fields and $G=\mathrm{Gal}(\mathbb{E}/\mathbb{F})$. Let $S \subseteq G$ (a sub{\it set} not necessarily a subgroup). We define $\mathbb{E}^S = \{ x \in \mathbb{E} \;|\; \sigma(x)=x \text{ for all } \sigma \in S\}$. This is the subfield of $\mathbb{E}$ {\it fixed} by $S$. 
\begin{enumerate}[(a)]
\item Show $\mathbb{E}^S$ is indeed a subfield of $\mathbb{E}$.
\item Suppose $T \subseteq S$. Show $\mathbb{E}^T \supseteq \mathbb{E}^S$.
\item {\bf [Grad.]} Let $H$ be a subgroup of $G$ and $\sigma \in G$. Show $\sigma H\sigma^{-1}$ is a subgroup of $G$. Also, show $\mathbb{E}^{\sigma H\sigma^{-1}}=\sigma(\mathbb{E}^H)$.

\force \phantom{\bf [Grad.]} What can be said when $H \triangleleft G$?
%\item {\bf [Grad.]} Considering part (b). It is possible to have an extension and choice of $T \subseteq S$ such that $\mathbb{E}^S \supsetneq \mathbb{E}^T$?
\end{enumerate}
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