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\title{MATH0037 Galois Theory 2002, Sheet 9}
\author{Geoff Smith, http://www.bath.ac.uk/$\sim$masgcs}
\date{}
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\begin{enumerate}
\item Consider the field $M = \mathbb Q (\omega, \alpha)$ discussed
in question 1 of sheet 8.
\begin{enumerate}
\item[(a)] List all 6 subgroups of $G = \mbox{Gal}_{\mathbb Q}M$.
\item[(b)] For each subgroup $H$ in the list you found in part (a),
describe the field $H^\dagger = \{ m \mid m \in M, (m)h = m \forall h \in H\}$
by exhibiting an element $\alpha_H \in H^\dagger$ such that 
$H^\dagger = \mathbb Q(\alpha_H)$. {\em Perhaps I should be writing
star instead of dagger. I have forgotten which notation was used in
lectures.}
\end{enumerate}
\item Suppose that $K \leq L$ is a field extension
and that $L$ is a splitting field for
some polynomial $f \in K[X]$. Prove that there are only finitely
many fields $T$ such that $K \leq T \leq L$. {\em You may assume the
truth of the fundamental theorem of Galois theory}. 
\item Suppose that $K \leq L$ is a field extension of finite degree.
Prove that there are only finitely
many fields $T$ such that $K \leq T \leq L$. {\em You may use the result
stated in the previous question}
\item Let $f = X^n -1 \in \mathbb Q[X]$. Let $\omega = e^{2\pi i/n} \in
\mathbb C$. Show that $Q(\omega)$ is a splitting field for $f$.
By considering what happens to $\omega$ under the action of elements
of $G = \mbox{Gal}_{\mathbb Q} Q(\omega)$, show that $G$ is
an abelian group.
\end{enumerate}


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