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\title{Math 421-621\\Spring 2013\\Homework 10}
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\centerline{\it Due Wednesday April 24, 2013.}

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\noindent 1. (5 pt) Let $K\subseteq F$ be an extension of fields with $F$ algebraic over $K$ and $D$ an integral domain with $K\subseteq D\subseteq F$. Show that $D$ is a field.

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\noindent 2. (5 pt) Let $K\subseteq F$ be fields. If $u\in F$ is algebraic over $K$ of odd degree, then $K(u)=K(u^2)$.

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\noindent 3. (5 pt) Using the notation of the previous problem, show that if $u$ is transcendental over $K$, then $K(u^2)$ is a proper subfield of $K(u)$ that is isomorphic to $K(u)$.

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\noindent 4. Let $K\subseteq F$ be fields. We define $\text{Aut}_K(F)$ to be the group of automorphisms of $F$ that fix $K$. (This is called the Galois group of $F$ over $K$.) Compute the following Galois groups.
\begin{itemize}
\item[a)] (5 pt) $\text{Aut}_{\mathbb{Q}}(\mathbb{Q}(\sqrt{d})$ where $d$ is a square free integer.
\item[b)] (5 pt) $\text{Aut}_{\mathbb{Q}}(\mathbb{Q}(\sqrt[3]{2})$.
\item[c)] (5 pt) $\text{Aut}_{\mathbb{Q}}(\mathbb{Q}(\sqrt{2},\sqrt{3})$.
\item[d)] (5 pt) $\text{Aut}_{\mathbb{Q}}(\mathbb{R})$.
\end{itemize}



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