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\title{MA exam: algebra}
\date{21 April 2007}
\author{Steinberg and Hewitt}
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\section*{MA exam: algebra, 21 April 2007}
Please do four problems, including one from each of the three
sections. Give complete proofs --- do not just quote a theorem.
Please indicate clearly which four problems you want to be graded.
\subsection*{Part I: Group theory}
\begin{problem}
Let $G$ be a cyclic group of order $n$ and let $d$ be a positive
divisor of $n$.
\begin{itemize}
\item[$a$.] Show that $G$ has an element of order $d$.
\item[$b$.] Suppose that $x$ and $y$ are elements of $G$ of order
$d$. Prove that there is an integer $m$ which is relatively prime
to $n$ such that $y=x^m$.
\end{itemize}
\end{problem}
\begin{problem}
Let $\phi\:G\to H$ be a group homomorphism which is onto.
\begin{itemize}
\item[$a$.] Suppose that $A$ is a normal subgroup of $G$. Prove that
$\phi(A)$ is a normal subgroup of $H$.
\item[$b$.] Suppose that $A$ is a subgroup of $G$ with the property
that $\phi(A)$ is a normal subgroup of $H$. Is it true that $A$
must be a normal subgroup of $G$? Either prove this is true or else
provide a counterexample.
\end{itemize}
\end{problem}
\subsection*{Part II: Ring theory}
\begin{problem}
Let $p$ be a prime integer and set
\[
R = \Bigl\{\frac mn \Bigm\vert
\text{$m$ and $n$ are integers and $p$ does not divide $n$}
\Bigr\}.
\]
You may assume that $R$ is a ring under the usual operations on
fractions.
\begin{itemize}
\item[$a$.] Find all of the units of $R$.
\item[$b$.] Show that each nonzero element of $R$ has the form $up^k$
where $u$ is a unit of $R$ and $k\ge0$.
\item[$c$.] Show that each ideal of $R$ is a principal ideal.
\item[$d$.] Find all of the irreducible elements in $R$.
\end{itemize}
\end{problem}
\begin{problem}
$a$. Prove that a euclidean domain is a principal ideal domain.
\begin{itemize}
\item[$b$.] Is $\R[x,y]$ a euclidean domain? (Here, $\R$ denotes the
field of real numbers.) Either prove that it is or else provide a
detailed explanation of why it is not.
\end{itemize}
\end{problem}
\subsection*{Part III: Linear algebra}
\begin{problem}
Let $V$ be the set of all $n\x n$ matrices over a field $F$. Let
$I$ denote the identity matrix. You may assume that $V$ is a vector
space over $F$, under the usual matrix operations.
\begin{itemize}
\item[$a$.] Give a basis for $V$.
\item[$b$.] Let $A\in V$. Prove that there is a positive integer $m$
and scalars $\alpha_0,\ldots,\alpha_{m-1}\in F$ such that
\[
A^m =
\alpha_0 I + \alpha_1 A + \alpha_2 A^2 + \cdots + \alpha_{m-1} A^{m-1}.
\]
\item[$c$.] Let $A\in V$. Show that $A$ is invertible if and only if
$A$ satisfies an equation as above with $\alpha_0\ne0$.
\end{itemize}
\end{problem}
\begin{problem}
Let $A$ be the $3\x3$ real matrix
\[
\begin{bmatrix}
1& 2& -3\\
1& 0& -1\\
1& 1& -2
\end{bmatrix}
\]
\begin{itemize}
\item[$a$.] Find the characteristic equation of $A$ and the
eigenvalues of $A$.
\item[$b$.] Find a basis for each eigenspace of $A$.
\item[$c$.] Is $A$ diagonalizable? If it is, diagonalize it. If not,
explain why not.
\end{itemize}
\end{problem}
\end{document}