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\title{MATH0037 Galois Theory 2002, Sheet 8}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
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\maketitle
\begin{enumerate}
\item Let $\alpha = \sqrt[3] 2$ and $\omega = e^{2\pi i/3}$.
Let $K = \mathbb Q[\alpha],\ L = \mathbb Q[\omega]$
and $M = K[\omega] = L[\alpha]$.
Note that $M$ is the smallest subfield of $\mathbb C$ which contains
both $\omega$ and $\alpha$.
\begin{enumerate}
\item[(a)] Prove that $|K:\mathbb Q| = 3$ and $|L: \mathbb Q| = 2$.
\item[(b)] Observe that $\omega \not \in \mathbb R$ so $\omega \not \in
K$. Deduce that $|M:K| = 2$ and so $|M:\mathbb Q| = 6$.
\item[(c)] Suppose that $\beta  \in \mathbb C$ is any root of
$X^3 -2$. Show that $L[\beta] = M$. 
\item Suppose that $\beta  \in \mathbb C$ is any root of
$X^3 -2$. There is a natural isomorphism $L[X]/(X^3-2) \rightarrow L[\beta]$.
Let $\beta_1, \beta_2$ be two distinct roots of
$X^3 -2$ in $\mathbb C$. Deduce that there is an element $g$ in 
Gal$_L M$ such that $(\beta_1) g = \beta_2$. Show that $g$ has order 3. 
\item[(d)] Show that complex conjugation induces an element $c$ in 
Gal$_K M$ of order $2$.
\item[(e)] Deduce that Gal$_{\mathbb Q} M$ contains an element
of order 2 and an element of order 3. 
\item[(f)] NOtice that $\Omega = \{ \alpha, \omega \alpha, \omega^2 \alpha\}$
is the set of complex roots of $X^3 -2$. Let $S_\Omega$ denote the
group of bijections from $\Omega$ to $\Omega$ (under composition 
of maps). Suppose that 
$h \in $ Gal$_{\mathbb Q} M$. Show that $(x)h \in \Omega$ for 
all $x \in \Omega$, and deduce that the application of $h$
induces an element $\widehat h \in S_\Omega$. Show that
the map $\eta:$ Gal$_{\mathbb Q} M \rightarrow S_\Omega$
defined by $h \mapsto \widehat h$ is a group homomorphism.
\item[(g)] Show that the homomorphism $\eta$ has trivial kernel,
so that $\eta$ is a monomorphism. 
\item[(h)] Use the fact that Gal$_{\mathbb Q} M$ contains elements of
order 2 and 3 to decude that $\eta$ is surjective, and is therefore
an isomorphism.
\end{enumerate}

\item Let $K$ denote the smallest subfield of $\mathbb C$ which contains
both $\sqrt 2$ and $\sqrt 3$. Show that $|K : \mathbb Q| = 4$.
By echoing the method of the previous paragraph, show that
Gal$_{\mathbb Q} K$ is a group of size 4. {\em (It is recommended
to use a new $\Omega = \{ \sqrt 2, -\sqrt 2, \sqrt 3, -\sqrt 3\}$
to replace the old $\Omega$.)}

\end{enumerate}


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