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{\bf University of Toledo Department of Mathematics\
Ph.D. Qualifying Exam in Algebra\\ Jan 26, 2008}
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\noindent \textbf{Instructions:} Please do {\em six} problems, including {\em at least one problem from each of the three sections}. Give complete proofs.\
If you attempt more than six problems, indicate clearly which six problems you would like graded. You have three hours.\
\section{Groups}
\begin{enumerate}
\item Prove that a group of order 825 is solvable.\\
\item Let $G$ be a group of order 2008. Prove that $G$ contains an abelian subgroup of index 2.\\
\item Prove that there is no simple group of order 90.\\
\item Let $H$ and $K$ be subgroups of a finite group $G$. Suppose that $|G:H| = p$, where $p$ is a prime and $p$ is strictly smaller than every prime divisor of $|K|$. Prove that $K$ is a subgroup of $H$.\\
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\section{Fields}
\item Let $K \subseteq F$ be a field extension. If $0 \ne b \in F$, show that $b$ is algebraic over $K$ if and only if $b^{-1}$ is a polynomial in $b$ with coefficients from $K$.\\
\item Find the Galois group of the splitting field over $\mathbb{Q}$ (the rational numbers) of the polynomial $x^4 - 14x^2 + 9$.\\
\item Let $E/K$ be a field extension of degree $p$, where $p$ is a prime. Suppose $f (x) \in K[x]$ is an irreducible polynomial which has more than one root in $E$. Prove that $f (x)$ splits in $E$.\\
\item If $F$ is a {\em finite} field, show that each element of $F$ is a sum of two squares. ({\em Hint:} Consider the set of squares in $F$ and the map $x \mapsto x^2$.)\\
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\section{Rings and Modules}
\item Let $R$ be a non-zero commutative ring and let $M$ be a non-zero torsion-free left $R$-module. ($M$ is said to be {\em torsion-free} if for any non-zero $m \in M$, if $r \in R$ such that $rm = 0$ then $r = 0$.) Show that there is an $R$-monomorphism from $M$ into a vector space over a field. ({\em Hint:} What if $M = R$?)\\
\item Suppose that $P_1$, $P_2$ and $P_3$ are ideals of a commutative ring $R$ and that $P_1$ is a prime ideal. If $S$ is a subring of $R$ and $S \subseteq P_1 \cup P_2 \cup P_3$, show that $S \subseteq P_i$ for some $i$.\\
\item Let $R$ be a non-zero ring with the property that each ascending sequence of ideals in $R$ is constant after finitely many terms.
(a) If $f: R \rightarrow R$ is a surjective ring homomorphism, prove that $f$ is an isomorphism.
(b) Show that the rings $R$ and $R[x]$ are not isomorphic.
(c) Give an example to show that (b) can fail if the condition on ascending sequences of ideals is not satisfied.\\
%\item A nonzero right module over the ring $R$ is {\em compressible} if it can be embedded (as an $R$- module) into each of its nonzero submodules.
%\begin{enumerate}
%\item Let $R$ be a commutative ring with identity and $I$ an ideal of $R$ other than $R$ itself. Show that $R/I$ is a compressible $R$-module if and only if $I$ is a prime ideal of $R$.
%\item Show that the following two conditions are equivalent for the compressible right $R$-module $M$:
%(i) There is no $R$-monomorphism from $M$ into a factor module $M/N$ with $N$ nonzero.
%
%(ii) Each $R$-homomorphism from a submodule of $M$ into $M$ is a monomorphism.
%
%\item Give an example of a module which satisfies the conditions in (b) and which has at least three distinct submodules.\\
%\end{enumerate}
\item Let $R$ be a ring and let $V$ be a right $R$-module. Assume that $M_1, M_2, \ldots, M_n$ are finitely many $R$-submodules of $V$ such that $M_1 \cap M_2 \cap \ldots \cap M_n = 0$, and let $W$ be the (external) direct sum $W = V/M_1 \oplus V/M_2 \oplus \ldots \oplus V/M_n$.
\begin{enumerate}
\item Show that $V$ is isomorphic to an $R$-submodule of $W$.
\item Suppose in addition that the modules $V/M_i$ are simple and pairwise nonisomorphic. Prove that $V$ is isomorphic to $W$.
\end{enumerate}
\end{enumerate}
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