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\begin{document}

\title{Math 81/111: Rings and Fields \\ Homework \#4}
\date{28 January 2015; due Thursday, 5 February 2015}

\maketitle

\begin{problab}{4.1}
Consider the polynomial $f(X)=X^4-2X^2-2$.  By the quadratic formula, the roots of $f$ are
\[ \pm \sqrt{1 \pm \sqrt{3}}. \]
\begin{enumalph}
\item Let $K_1=\Q(\sqrt{1+\sqrt{3}})$ and $K_2 = \Q(\sqrt{1-\sqrt{3}})$.  Show that $K_1 \neq K_2$ (as subfields of $\C$) and $K_1 \cap K_2 = \Q(\sqrt{3})$.  
\item Let $F=K_1 \cap K_2$.  Show that $K_1$, $K_2$, and $K_1K_2$ are Galois over $F$.  
\item Show that $\Gal(K_1K_2/F)$ is the Klein $4$-group.  List the automorphisms in this group and their action on the roots in (a).
\item Let $L$ be a splitting field of $f$.  Show that $\Gal(L/\Q) \cong D_8$ is isomorphic to the dihedral group of order $8$.  Make a nice field diagram and corresponding subgroup diagram showing the Galois correspondence for each subfield and corresponding subgroup.
\end{enumalph}
\end{problab}

\begin{problab}{4.2}
Let $K=\Q(\sqrt{2+\sqrt{2}})$.  Show that $K$ is Galois over $\Q$ and that $\Gal(K/\Q)$ is a cyclic group.
\end{problab}

\begin{problab}{4.3 (M3-2)}
Let $p$ be an odd prime, and let $\zeta=\zeta_p \in \C$ be a primitive $p$th root of unity and let $K=\Q(\zeta)$.
\begin{enumalph}
\item Show that $K$ is Galois and that the map
\begin{align*} 
(\Z/p\Z)^\times &\to \Gal(K/\Q)=G \\
a &\mapsto (\zeta \mapsto \zeta^a) 
\end{align*}
is an isomorphism of groups.
\end{enumalph}

\noindent Now let $H \leq G$ be the subgroup of index $2$ in $G$.  Let
\[ \alpha=\sum_{\sigma \in H} \sigma(\zeta), \quad \beta = \sum_{\sigma \in G \setminus H} \sigma(\zeta). \]
\begin{enumerate}
\item[(b)] Show that $\alpha,\beta \in K^H$, and that if $\sigma \in G \setminus H$, then $\sigma$ swaps $\alpha$ and $\beta$.  Conclude that $\alpha,\beta$ are the roots of the polynomial $X^2+X+\alpha\beta \in \Q[X]$.
\item[(c)] Show that 
\[ 4\alpha\beta = 
\begin{cases}
p-1, & p \equiv 1 \pmod{4}; \\
p+1, & p \equiv 3 \pmod{4}.
\end{cases} \]
and conclude that the fixed field $K^H=\Q(\sqrt{ \pm p})$, accordingly.  \emph{[Hint: Make sense of the `solution' to \textup{M3-2} on page 128--129 and write it out fully.]} 
\end{enumerate}
\end{problab}

\begin{problab}{4.4 (M3-4)}
Let $K/F$ be a finite Galois extension with Galois group $G$ and let $M=K^H$ with $H \leq G$.  Show that $\Aut_F(M)=N/H$ where $N=N_G(H)$ is the normalizer of $H$ in $G$.
\end{problab}


\end{document}