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\begin{document}
\section*{Qualifying exam in algebra, April 2005}
Do four of the following problems. Please give complete solutions.
Two half solutions count less than one complete solution.
Indicate clearly which four problems you wish to have graded.
\begin{problem}
Let $K=\GF{4}$ and $L=\GF{64}$, where $\GF{q}$ denotes the field with
$q$ elements.
\begin{itemize}
\item[a.]
Show that $K$ is (isomorphic to) a subfield of $L$.
\item[b.]
Find a polynomial over $K$ whose splitting field is $L$.
\item[c.]
Determine the Galois group of $L/K$.
\end{itemize}
\end{problem}
\begin{problem}
Suppose that $G$ is a group.
\begin{itemize}
\item[a.]
For $g\in G$ let $c_g$ denote the map
$x\mapsto x^g=g^{-1}xg$. Show that the map $G\to\Aut G$ given by
$g\mapsto c_g$ is a group homomorphism, and identify its kernel.
\item[b.]
A group is said to be \textit{complete} if the map in part (a) is an
isomorphism. Suppose that $N$ is a normal subgroup of $G$,
and that $N$ is complete. Show that $G\cong N\x G/N$.
\end{itemize}
\end{problem}
\begin{problem}
The ring $R$ is \textit{prime} if whenever $A$ and $B$ are ideals of
$R$ with $AB=0$ then either $A=0$ or $B=0$.
\begin{itemize}
\item[a.]
Show that $R$ is prime if and only if whenever $a,b\in R$ such that
$aRb=0$ then either $a=0$ or $b=0$.
\item[b.]
Give an example of a prime ring that is not a domain.
Verify both that it is prime and that it is not a domain.
\end{itemize}
\end{problem}
\begin{problem}
Let $R$ be a commutative domain with identity and let $S$ be a be a
subset with the following properties: $0\notin S$; $1\in S$; if
$a,b\in S$ then $ab\in S$. Let $Q$ denote the field of quotients of
$R$, and set
\[
R_S = \{as^{-1}\in Q : a\in R, s\in S\}.
\]
You may use without proof that $R_S$ is a subring of $Q$ which
contains $R$.
\begin{itemize}
\item[a.]
Show that if $R$ is a PID then so is $R_S$.
\item[b.]
Suppose $R=\Z$, $p$ is a prime, and
\[
S = \{n\in\Z : \text{$p$ does not divide $n$}\}.
\]
Find all ideals of $R_S$.
\end{itemize}
\end{problem}
\begin{problem}
A permutation $\alpha\in\Sym n$ is \textit{regular} if \textit{all} of
the cycles in the cycle decomposition of $\alpha$ have the same
length. Note that the only regular permutation which fixes a point is
the identity.
\begin{itemize}
\item[a.]
Prove that $\alpha$ is regular if and only if $\alpha$ is a power of
an $n$-cycle.
\item[b.]
Let $G$ be a finite group and let $a\in G$. Prove that the
permutation $\alpha\:x\mapsto ax$ on $G$ is regular.
\end{itemize}
\end{problem}
\begin{problem}
\newcommand\sigmabar{\overline{\sigma}}
\newcommand\taubar{\overline{\tau}} If $\sigma\:F\to F_1$ is an
isomorphism of fields, let $\sigmabar\:F[x]\to F_1[x]$ be defined by
the rule
\[
\sigmabar\left(\sum a_i x^i\right) = \sum \sigma(a_i) x^i.
\]
\begin{itemize}
\item[a.]
Show that $\sigmabar$ is an isomorphism.
\item[b.]
Suppose that $f(x)\in F(x)$ and let $f_1(x)=\sigmabar(f_1(x))$. Let
$L$ be a splitting field of $f(x)$ over $F$ and let $L_1$ be a
splitting field of $f_1(x)$ over $F_1[x]$. Show that there is an
isomorphism $\tau\:L\to L_1$ such that $\tau(a)=\sigma(a)$ for every
$a\in F$.
\item[b.]
Give an example to show that there may be more than one choice for
$\tau$.
\end{itemize}
\end{problem}
\end{document}
\begin{problem}
Let $A$ be an algebra of matrices, of finite degree $d$, over the
commutative field $K$. Suppose that $A$ is closed under transpose ---
that is, if $x\in A$ then $x^T\in A$. Show that $A$ is semisimple.
\end{problem}
\begin{problem}
Prove or give a counterexample: If $G$ is a group, and $G=HK$, where
$H$ and $K$ are solvable subgroups, then $G$ is solvable.
\end{problem}
\begin{problem}
Prove or give a counterexample: If $G$ is a group, and $G=HK$, where
$H$ and $K$ are solvable subgroups, and if $H$ is normal then $G$ is
solvable.
\end{problem}
\begin{problem}
Prove or give a counterexample: If $G$ is a group, and $G=HK$, where
$H$ and $K$ are nilpotent subgroups, and if $H$ is normal then $G$ is
nilpotent.
\end{problem}
\begin{problem}
Prove or give a counterexample: If $K$ is a commutative field in which
every element has a square root, then every finite extension of $K$
has odd degree.
\end{problem}
\begin{problem}
Suppose $f$ is a quintic polynomial, irreducible over
$\Q$, whose graph over $\R$ cuts the $x$-axis in three places.
Identify the Galois group of $f$.
\end{problem}
\begin{problem}
A subgroup $H$ of a group $G$ is said to be \textit{pronormal} in case
it its normalizer is self-normalizing --- that is, in case
$N_G(N_G(H))=N_G(H)$. Show that the Sylow subgroups of a finite group
are pronormal.
\end{problem}
\begin{problem}
A \textit{character} of a finite abelian group $A$ is a
homomorphism $A\to\C^\x$, the multiplicative groups of nonzero complex
numbers. The collection of all characters of $A$ is an abelian group,
under pointwise operations. It is denoted $A^\vee$. In this problem,
$A$ always denotes finite abelian group.
\begin{itemize}
\item[a.]
Show that
\[
\frac1{\abs{A}} \sum_{x\in A} \alpha(x) =
\begin{cases}
1, & \text{if $\alpha\equiv1$}; \\
0, & \text{otherwise}.
\end{cases}
\]
\item[b.]
Let $\C^A$ be the vector space of all complex-valued functions on
$A$. Define an inner product on this space by the following rule:
\[
\langle f\mid g\rangle=
\frac1{\abs{A}} \sum_{x\in A} f(x)\overline{g(x)}.
\]
Show that $A^\vee$ is an orthonormal basis of $\C^A$, with
respect to the inner product above.
\end{itemize}
\end{problem}
\begin{problem}
Let $G=\GL2{\GF{11}}$, the group of $2\x2$ invertible matrices over the
field with $11$ elements. Determine the conjugacy classes of $G$. In
particular, find the size and a representative of each conjugacy class.
\end{problem}
\end{document}