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{\bf Preliminary Draft of Algebra Ph.D. Qualifying Exam- January 27, 2007}
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\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.
\section{Section I}
\begin{enumerate}
\item Let $G$ be a group with exactly three subgroups (including the
trivial subgroup and $G$ itself).
\begin{enumerate}
\item Prove that $G$ is finite and cyclic.
\item Prove that the order of $G$ is $p^2$ for some prime $p$.\\
\end{enumerate}
\item Let $G$ be a non-abelian $p$-group of order $p^3$, where $p$
is a prime number. Let $Z(G)$ be the center of $G$ and $G'$ be its
commutator subgroup. \begin{enumerate}
\item Show that $Z(G) = G'$ and that this is the unique normal
subgroup of $G$ of order $p$.
\item Determine the number of distinct conjugacy classes of $G$.\\
\end{enumerate}
\item Let $G$ be a finite group having exactly $n$ Sylow
$p$-subgroups for some prime $p$. Show that there exists a subgroup
$H$ of the symmetric group $S_n$ of degree $n$ that also has exactly
$n$ Sylow $p$-subgroups.\\
\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}
\pagebreak
\section{Section II}
\item Prove or disprove: There exist two non-isomorphic rings, each with 9 elements, whose additive
groups are isomorphic.\\
\item Prove that the group of all automorphisms of the field
$\mathbb{R}$ of real numbers is trivial.\\
\item Let $f(x) = x^5 - 9x + 3 \in \mathbb{Q}[x]$. Determine the
Galois group of $f(x)$ over $\mathbb{Q}$.\\
\item
\begin{enumerate}
\item Prove that if $R$ is a commutative ring with one. Prove that
every maximal ideal of $R$ is a prime ideal.
\item Show that the ideal $(3,x)$ of $\mathbb{Z}[x]$ generated by 3
and $x$ is a maximal ideal of $\mathbb{Z}[x]$.
\item Find a prime ideal of $\mathbb{Z}[x]$ that is {\em not}
maximal.\\
\end{enumerate}
\item Let $K$ be the field obtained by adjoining to the rational
numbers $\mathbb{Q}$ all complex cube roots of $2$.
\begin{enumerate}
\item Determine the degree $|K:\mathbb{Q}|$.
\item Determine the Galois group of the extension $K/\mathbb{Q}$.
\item Determine all subfields of $K$.\\
\end{enumerate}
\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
that the degree $|E:K|$ is equal to the characteristic of $F$.\\
\end{enumerate}
\item Let $\alpha$ be a non-zero real number and suppose that
$\alpha^n \in \mathbb{Q}$, the rational numbers, for some integer
$n$. Let $g(x)$ be the minimal (monic) p[olynomial of $\alpha$ over
$\mathbb{Q}$ and let $\deg g = m$.
\begin{enumerate}
\item Show that $g(0) = \pm \alpha^m$.
\item Prove that $g(x) = x^m - b$ for some $b \in \mathbb{Q}$.
\item Show that $m$ divides $n$.\\
\end{enumerate}
\pagebreak
\section{Section III}
\item Let $A$ be an $n \times n$ matrix over a field $K$ and assume that the characteristic polynomial of $A$ has
distinct roots in the algebraic closure of $K$. Prove that any two
$n \times n$ matrices that commute with $A$ must commute with each
other.\\
\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 $V$ be a vector space over an algebraically closed field
$K$ and let $T: V \rightarrow V$ be a linear operator on $V$. Let
$I: V \rightarrow V$ denote the identity operator. Show that $V$ has
a basis consisting of eigenvectors of $T$ if and only if the kernel
of $(\lambda I - T)^2$ is equal to the kernel of $\lambda I - T$ for
all $\lambda \in K$.\\
\item Let $\mathbb{Z}[i]$ denote the Gaussian integers. Prove or
disprove: $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{R} \cong
\mathbb{C}$.\\
\end{enumerate}
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