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\title{Math 721\\Spring 2011\\Homework 4}
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\centerline{\it Due Friday March 25, 2011.}

\centerline{}

\noindent 1. Find the canonical forms (the rational canonical form, primary rational canonical form and Jordan canonical form if possible) for the following matrices over $\mathbb{Q}$:
\begin{itemize}
\item[a)] (5 pt)
$\left[
\begin {array}{ccccc}
\ \ 3&\ \ 1&\ \ 0&\ \ 1&\ \ 1\\

\ \ 0&\ \ 3&\ \ 0&-1&\ \ 0\\

\ \ 0&-2&\ \ 4&\ \ 2&\ \ 0\\

\ \ 0&-1&\ \ 0&\ \ 3&\ \ 0\\

\ \ 1&-1&\ \ 0&-1&\ \ 3
\end {array}
\right]$

\item[b)] (5 pt) 
$\left[
\begin {array}{ccccc}
\ -1&\ \ 1&\ \ 0&\ \ 0&\ \ 0\\

\ \ 1&-2&-1&\ \ 0&\ \ 0\\

\ -5&\ \ 8&\ \ 3&\ \ 0&\ \ 0\\

\ \ 15&-31&-9&\ \ 3&\ \ 1\\

\ \ -40&82&\ \ 21&-9&\ \ -3
\end {array}
\right]$

\end{itemize}

\centerline{}

\noindent 2. For the following fields, $F$, find the group of $\mathbb{Q}-$automorphisms of $F$. 
\begin{itemize}
\item[a)] (5 pt) $F=\mathbb{Q}(i)$
\item[b)] (5 pt) $F=\mathbb{Q}(\sqrt[3]{2})$
\item[c)] (5 pt) $F=\mathbb{R}$.
\end{itemize}

\centerline{}

\noindent 3. (5 pt) Let $K\subseteq D\subseteq F$ with $K, F$ fields with $F$ algebraic over $K$ and $D$ an integral domain. Show that $D$ is a field.




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