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\title{MATH0037 Galois Theory 2000}
\author{GCS -- Sheet 6}
\date{}
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\maketitle

\begin{enumerate}
\item Suppose that the quadratic $f \in \mathbb Q[X]$ has roots $\alpha, \beta
\in \mathbb C$. Suppose furthermore that $f \in \mathbb Q[X]$
is irreducible. Let $K = \mathbb Q(\alpha, \beta)$.
\begin{enumerate}
\item Notice that $\mathbb Q(\alpha) \simeq \mathbb Q[X]/(f)$.
Deduce that the fields $\mathbb Q(\alpha), \mathbb Q(\beta)$ are isomorphic
via an isomorphism which fixes rational numbers and sends
$\alpha$ to $\beta$.
\item Prove that $K = \mathbb Q(\alpha) = \mathbb Q(\beta)$.
\item Prove that $\hbox{Gal}_{\mathbb Q}K$ is a cyclic
group of order 2.
\end{enumerate}
\item Suppose that $\mathbb Q \leq F$ is a field extension
and that $f: F \rightarrow F$ is an isomorphism of fields.
Prove that $f$ fixes each element of $\mathbb Q$.
\item Let $\lambda = e^{2\pi i/5} \in \mathbb C$.
\begin{enumerate}
\item Let $f(X) = X^4 + X^3 + X^2 + X + 1$. Show that 
$\lambda$ is a root of $f$.
\item Notice that $f(X) = [X^5 -1]/[X-1]$. Let $h(X) = f(X+1)$.
Use Eisenstein's criterion to deduce that $h(X) \in \mathbb Q[X]$ is
irreducible.
\item Deduce that $f(X) \in \mathbb Q[X]$ is
irreducible.
\item Suppose that $\theta, \psi \in \mathbb C$
are both fifth roots of 1 and neither of them is 1.
Prove that there is an isomorphism between the
fields $\mathbb Q(\theta)$ and $\mathbb Q(\psi)$
which sends $\theta$ to $\psi$ and fixes each rational number.
\item Let $K$ be the field $\mathbb Q(\lambda)$. 
What is the Galois group $G$ of the field extension
$\mathbb Q \leq K$? (i.e. how many elements does $G$ have and 
what are they?) 
\end{enumerate}
\end{enumerate}
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