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{\bf University of Toledo Algebra Ph.D. Qualifying Exam\\ April 21, 2007}
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\vspace{0.3in}
\noindent \textbf{Instructions:} The exam is divided into three
sections. Please choose exactly three problems from each section.
Clearly indicate which three you would like graded. You have three
hours.\\
%$\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ denote, respectively,
%the rational numbers, the real numbers and the complex numbers.
\vspace{0.3in}
\section{Section I}
\begin{enumerate}
%\item Describe all the groups of order 12.\\
\item
\begin{enumerate}
\item Find the Sylow-3 subgroups of the symmetric group $S_4$.
\item Let $f$ be an automorphism of $S_4$. Show that $f$ permutes the Sylow-3 subgroups and that if $f$ fixes them all then $f$ is the trivial automorphism. Conclude that $|\operatorname{Aut}S_4| \leq 24$.
\item Show that $S_4$ has 24 inner automorphisms, and thus $|\operatorname{Aut}S_4| \cong S_4$ and $S_4$ has no outer automorphisms.\\
\end{enumerate}
%\item Let $k$ be a field and let $G = GL_n(k)$ be the general linear
%group of all invertible $n \times n$ matrices over $k$. Let $D$ be
%the subgroup of diagonal matrices and $N = N_G(D)$ be the normalizer
%of $D$ in $G$. Determine (up to isomorphism) the quotient group
%$N/D$.\\
%
%\item Suppose that $G$ is the (internal) direct product of subgroups $S$ and $T$. Let $H$ be a subgroup of $G$ such that $SH = G = TH$\\
%\begin{enumerate}
%\item Prove that $S \cap H$ and $T \cap H$ are normal subgroups of $G$.
%\item If $S \cap H = 1 = T \cap H$, prove that $S$ and $T$ are isomorphic.
%\item If $S \cap H = 1 = T \cap H$, prove that $G$ is abelian.\\
%\end{enumerate}
%\item Suppose $K \unlhd G$ are finite groups. \\
%\begin{enumerate}
%\item Define what it means for $G$ to \emph{split} over $K$.
%\item Suppose $G/K$ is a $p$-group and let $P$ be a Sylow $p$-subgroup of $G$. Prove that $G$ splits over $K$ if and only if $P$ splits over $P \cap K$.\\
%\end{enumerate}
\item Let $G$ be a finite group, $H \leq G$, $N \unlhd G$. Prove that if $|H|$ and $|G:N|$ are relatively prime then $H \leq N$.\\
%\item Let $P$ be a Sylow $p$-subgroup of $H$ and let $H$ be a subgroup of $K$. If $P \unlhd H$ and $H \unlhd K$, prove that $P \unlhd K$. Deduce that normalizers of Sylow subgroups are self-normalizing, i.e. if $P$ is a Sylow subgroup of $G$ that $N_G(N_G(P))=N_G(P)$.\\
\item Let $p$ and $q$ be distinct primes and suppose that $G$ is a finite group having exactly $p+1$ Sylow $p$-subgroups and $q + 1$ Sylow $q$-subgroups. Prove that there exist $P \in Syl_p(G)$ and $Q \in Syl_q(G)$ such that the subgroup generated by $P$ and $Q$ is $PQ = P \times Q$.\\
\item Let $x$ and $y$ be elements of a finite $p$-group $P$ and let $z = [x,y]$ be the commutator $x^{-1}y^{-1}xy$ of $x$ and $y$. suppose that $x$ lies in every normal subgroup of $P$ that contains $z$. Prove that $x = 1$.\\
\item Let $G$ be a group and let $N$ be a normal subgroup of $G$.\\
\begin{enumerate}
\item If $G/N$ is a {\em free} group, prove that there is a subgroup $K$ of $G$ such that $G = NK$ and $N \cap K = 1$.
\item Show that the conclusion in part (a) is false if $G/N$ is not assumed to be free.\\
\end{enumerate}
%\item Let $G$ be a group of order 105.\\
%\begin{enumerate}
%\item Prove that $G$ contains a normal subgroup of order 35.
%\item Prove that, up to isomorphism, there are exactly two possibilities for $G$.\\
%\end{enumerate}
%\item Let $C_2$ denote the group of order 2 and let $D_m$ denote the dihedral group of order $2m$. Let $n$ be an odd natural number.\\
%
%\begin{enumerate}
%\item Prove that $D_{2n} \cong D_n \times C_2$.
%\item Prove that $D_{4n}$ and $D_{2n} \times C_2$ are {\em not} isomorphic.
%\end{enumerate}
\newpage
\section{Section II}
\item Prove that the group of all automorphisms of the field
$\mathbb{R}$ of real numbers is trivial.\\
%\item Give a polynomial in $\mathbb{Q}[x]$ which is not solvable by radicals. Prove that it is not solvable by radicals.\\
%
%
%
%\item Let $E$ be a finite Galois extension of the field $F$ with
%Galois group $Gal(K/F) = G$. Let $K$ be an intermediate field and
%let $H$ be the subgroup of $G$ consisting of the elements of $G$
%that fix all elements of $K$. Show that the subgroup of $G$
%consisting of all $\sigma \in G$ for which $\sigma(K) = K$ is the
%normalizer $N_G(H)$ of $H$ in $G$.\\
%
%\item Let $F$ be a field, $f(x) \in F[x]$ be an irreducible
%polynomial and $E$ be a splitting field for $f(x)$ over $F$. Assume
%there exists and element $\alpha \in E$ such that both $\alpha$ and
%$\alpha + 1$ are roots of $f(x)$.
%\begin{enumerate}
%\item Show that the characteristic of $F$ is not zero.
%\item Prove that there exists a field $K$ between $F$ and $E$ such ??????\\
%\end{enumerate}
%
%\item
%\begin{enumerate}
%\item Put the following matrix into rational canonical form:
%
%$$\left(%
%\begin{array}{cccccc}
% 2 & 1 & 0 & 0 & 0 & 0 \\
% 0 & 2 & 0 & 0 & 0 & 0 \\
% 0 & 0 & 2 & 1 & 0 & 0 \\
% 0 & 0 & 0 & 2 & 0 & 0 \\
% 0 & 0 & 0 & 0 & 3 & 0 \\
% 0 & 0 & 0 & 0 & 0 & 5 \\
%\end{array}%
%\right)$$
%\item Prove that an $n \times n$ matrix with complex entries satisfying $A^3=A$ can be diagonalized. Is the same statement try over any field $F$?\\
%\end{enumerate}
\item Determine the Galois group of $f(x)=x^4-2 \in \mathbb{Q}[x]$. Illustrate explicitly the lattice of subgroups and the corresponding lattice of subfields under the fundamental theorem of Galois Theory.\\
%\item If $K$ is a field, $f(x) \in K[x]$ is irreducible of degree $d \ge 1$ and if $L/K$ is a finite extension of degree $n$ with $\gcd(n,d) = 1$, prove that $f(x)$ is irreducible in $L[x]$.\\
\item We say a field extension $K/F$ is {\em cyclic} if it is Galois and the Galois group is cyclic.\\
(a) Let $F$ be a field of characteristic 0 and assume that $K/F$ is cyclic of degree $|K:F| = n$. If $d$ is any divisor of $n$, show that there is a unique intermediate field $L$ such that $L/F$ is cyclic of degree $d$.\\
(b) Assume (a special case of) Dedekind's theorem that, for any natural number $n$, there are infinitely many primes of the form $kn + 1, k \in \mathbb{Z}$. Prove that for any natural number $n$, there is an extension of the field $\mathbb{Q}$ of rational numbers that is cyclic of degree $n$.\\
\item Let $F$ and $K$ be fields with $F \subseteq K$ and assume that the extension $K/F$ is algebraic. If $\sigma: K \rightarrow K$ is a ring homomorphism that fixes each element of $F$, prove that $\sigma$ is an $F$-isomorphism.\\
\item Let $f$ be an irreducible polynomial of degree 6 over a field $F$. Let $K$ be an extension field of $F$ with $|K:F| = 2$. If $f$ is {\em reducible} over $K$, prove that it is the product of two irreducible cubic polynomials over $K$.
\pagebreak
\section{Section III}
%\item Let $F$ be an algebraically closed field of prime characteristic
%$p$ and let $V$ be an $F$-vector space of dimension exactly $p$.
%Suppose that $A$ and $B$ are $F$-linear operators on $V$ such that
%$AB-BA=B$. If $B$ is non-singular, prove that $V$ has a basis
%$\{v_1, v_2, \ldots, v_p \}$ of eigenvectors of $A$ such that $Bv_i
%= v_{i+1}$ for $1 \le i \le p-1$ and $Bv_p = \lambda v_1$ for some
%$\lambda$, $0 \ne \lambda \in F$.\\
\item Let $R$ be a commutative ring with identity and $P$ a prime ideal.
\begin{enumerate}
\item Describe the construction of the localization of $R$ at $P$, denoted $R_P$.
\item Prove there is a 1-1 correspondence between prime ideals of $R$ which are contained in $P$ and prime ideals of $R_P$.
\item Prove that under this correspondence the ideal $P$ corresponds to the unique maximal ideal in $R_P$.
\item Prove this maximal ideal is exactly the set of non-units in $R_P$.\\
\end{enumerate}
\item Let $I$ be a principal ideal in an integral domain $R$. Prove that the $R$-module $I \otimes_R I$ has no nonzero torsion elements.\\
\item (a) Let $F$ be a field and let $A$ be an $n \times n$ matrix with entries in $F$. State a necessary and sufficient condition on the minimal polynomial of $A$ for $A$ to be diagonalizable over $F$.\\
\noindent (b) Let $F = \mathbb{C}$ be the field of complex numbers. If $A$ satisfies the equation $A^3 = -A$, show that $A$ is diagonalizable over $\mathbb{C}.$\\
\noindent (c) Let $F = \mathbb{R}$ be the field of real numbers. Given that $A$ satisfies the equation $A^3 = -A$ and given that $A$ is diagonalizable over $\mathbb{R}$, what is the strongest conclusion that can be drawn about $A$?\\
\item Let $F$ be a field. Construct, up to similarity, all linear transformations $T: F^6 \rightarrow F^6$ with minimal polynomial $m_T(x) = (x - 5)^2(x-6)^2$,\\
\item (a) Let $R$ be a ring and $M$ be an $R$-module. What does it mean for $M$ to be a {\em free} $R$-module?\\
\noindent (b) Let $\mathbb{Z}[\frac{1}{2}]$ denote the subring of $Q$ generated by $\mathbb{Z}$ and $\frac{1}{2}$. Prove or disprove: $\mathbb{Z}[\frac{1}{2}]$ is a free $\mathbb{Z}$-module.\\
\end{enumerate}
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