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\title{Math 728\\Fall 2004\\Final Exam}
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\centerline{\it Due Wednesday December 15, 2004.}

\centerline{}

\noindent 1. Suppose that $A$ is a matrix (over a field, $\mathbb{F}$) with characteristic polynomial $f(x)=x^6+2x^4-7x^2+4$.
\begin{itemize}
\item[a)] (5 pt) Suppose that the equation $y^2+1=0$ has a solution in $\mathbb{F}$. Find all possible rational canonical forms, primary rational canonical forms, and Jordan forms (if possible).
\item[b)] (5 pt) Suppose that the equation $y^2+1=0$ has no solution in $\mathbb{F}$. Find all possible rational canonical forms, primary rational canonical forms, and Jordan forms (if possible).  
\end{itemize}




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\noindent 2. (5 pt) Show that if $M$ is a nilpotent matrix over a field $\mathbb{F}$, then all the eigenvalues of $M$ are $0$. Use this to find all possible Jordan forms of a $5\times 5$ matrix over a field $\mathbb{F}$.

\centerline{}

\noindent 3. ({\it The Picard group}) Let $R$ be an integral domain with quotient field $K$. Let $\text{Pic}(R)$ denote the isomorphism classes of rank 1 (finitely generated) projective $R-$modules. For two isomorphism classes, $[P]$ and $[Q]$, we define multiplication via

\[
[P]\circ [Q]=[P\otimes_R Q].
\]

\begin{itemize}
\item[a)] (5 pt) Show that $\text{Pic}(R)$ forms a monoid with this multiplication.
\item[b)] (5 pt) If $P$ is a projective $R-$module, show that $P^*=\text{Hom}_R(P,R)$ is also a projective $R-$module.
\item[c)] (5 pt) Show that if $P$ is a (finitely generated) rank 1 projective $R-$module, then we can identify $P$ as a fractional invertible ideal of $R$ (hint: use the fact that $P$ is rank 1, tensor the injection $R\longrightarrow K$ with $P$; you may use the fact from class that a fractional ideal is invertible if and only if it is projective).
\item[d)] (5 pt) Conclude that $\text{Pic}(R)$ forms a group by showing that $[P]^{-1}=\text{Hom}_R(P,R)$.
\end{itemize}

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\noindent 4. (5 pt) Let $R$ be commutative with identity. Show that $K_0(R)$ is a direct summand of $K_0(R[x])$.
    

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\noindent 5. (5 pt) Let $R$ be a Euclidean domain. Show that if $M$ is an $n\times n$ matrix over $R$, then $R$ can be reduced to a matrix of the form
\begin{center}
\[
\left[
\begin {array}{ccccc}
\ \ \lambda_1&\ \ 0&\ \ 0&\ \ \cdots&\ \ 0\\

\ \ 0&\ \ \lambda_2&\ \ 0&\ \ \cdots&\ \ 0\\

\ \ 0&\ \ 0&\ \ \lambda_3&\ \ \cdots&\ \ 0\\

\ \ \vdots &\ \ \vdots&\ \ \vdots&\ \ \vdots&\ \ \vdots\\

\ \ 0&\ \ 0&\ \ 0&\ \ \cdots&\ \ \lambda_n
\end {array}
\right]
\]
\end{center}
\noindent with $\lambda_1\vert\lambda_2\vert\cdots\vert\lambda_n$ by using elementary row and column operations.

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