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\begin{document}

\begin{center}
{\bf\large  ALGEBRA TEST \#2}
\end{center}

\vspace{1in}

This exam is due Friday, March 23.
You are expected to complete three problems,
one from each group.
Clearly mark which problems are to be graded.

You may use your book, but you may not
communicate with others concerning
the exam.
In order to receive full credit your answer must be 
{\bf complete}, {\bf legible} and {\bf correct}. 

\vfill

\begin{center}
I have neither given nor received aid on this exam.
\end{center}

\vspace{.5in}

\begin{flushright}
{\bf Name:\underline{\hspace{2.5in}}}
\end{flushright}


\break

\noindent
{\bf Group 1.}

\medskip

1. Let $A$ be an $m\times m$ complex matrix. Show that if
$A^n = I$, then $A$ is similar to a diagonal matrix.

\vfill

2. Prove or disprove: the rational canonical form 
of a permutation matrix is a permutation matrix.

\vfill

\noindent
{\bf Group 2.}

\medskip

3. Find the rational canonical form and the Jordan 
canonical form of 

\[\left[\begin{array}{rrrr}
2 & -1 & \hphantom{-}1 & 1\\
0 & 1 & 0 & 0\\
-1 & 0 & 0 & -1\\
0 & 1 & 0 & 1
\end{array}\right]\]
\vfill

4. Let $\m m$ be a finitely generated module over a
commutative Artinian ring. Show that any 
injective homomorphism $\varphi:\m m\rightarrow \m m$
is an isomorphism.

\vfill

\noindent
{\bf Group 3.}

\medskip

5. Let $\m r$ be the subring of ${\BQ}$ whose elements
are $\displaystyle \left\{\frac{m}{n}\in {\BQ}\;|\;\mbox{\rm 
$n$ odd}\right\}$. $\m r$ is a PID. Describe the isomorphism types
of finitely generated torsion $\m r$-modules. For which modules
is it true that multiplication by $\displaystyle \frac{11}{43}$ 
is the identity function?

\vfill

6. Let $\vr V$ be the category of vector spaces over
the field ${\BF}$. Let $I: {\vr C}\rightarrow {\vr C}$
be the identity functor: $I(\m v) = \m v$, $I(\varphi) = \varphi$.
Let $F:{\vr C}\rightarrow {\vr C}$ be the squaring functor:
$F(\m v) = \m v\times \m v$, 
$F(\varphi) = \varphi\times \varphi$.\footnote{$(\varphi\times \varphi)(x,y) 
= (\varphi(x),\varphi(y))$.}
Show that addition
$\eta_{\m v}: \m v\times \m v\rightarrow \m v: (x,y)\mapsto x+y$
is a natural transformation from $F$ to $I$.

\vfill

\end{document}
