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\begin{document}

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

\vspace{1in}

This exam is due Wednesday, March 20.
Do {\bf four} of the problems.
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

1. Let $\m r$ be an integral domain with
field of fractions $\m q$, and let
$\m m$ be an $\m r$-module of rank $n$.
Show that $\m q\otimes_{\m r} \m m$ is a
$\m q$-vector space of dimension $n$.

\vfill

2. Let $V$ be a finite dimensional $\BF$-space. Show
that the function $\varphi\mapsto \varphi^*$
is an isomorphism from the $\BF$-algebra
${\rm End}_{\BF}(V)^{\rm op}$
to ${\rm End}_{\BF}(V^*)$.
Show that the map is not surjective if $V$
is infinite dimensional.

\vfill

3. Let $V$ be an $n$-dimensional $\BF$-space.
Our definition of the determinant as an
element of $\Lambda^n(V)$ depended on a choice
of basis for $V$. If we change the basis of $V$,
then will the values of the determinant change?
If so, how is the change in value computed?
%Is every alternating $n$-form on $V$ a determinant
%for some basis?

\vfill

4. Let $\m m$ be a module over an integral domain.
Show that $\m m/{\rm Tor}(\m m)$ is torsion free,
but may not be free. Show that if
$\m m/{\rm Tor}(\m m)$ is free, then 
$\m m$ is isomorphic to 
%$(\m m/{\rm Tor}(\m m))\oplus {\rm Tor}(\m m)$.
${\rm Tor}(\m m)\oplus (\m m/{\rm Tor}(\m m))$.

\vfill

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. Show that any
finitely generated torsion $\m r$-module is finite. 

\vfill

6. Let $\m a$ be a Noetherian algebra. Show that any 
surjective homomorphism $\varphi:\m a\rightarrow \m a$
is an isomorphism.

\vfill

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

\vfill

8. 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


\end{document}
