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{\bf Algebra Ph.D. Qualifying Exam- April 15, 2006}
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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 Classify completely the possible isomorphism type of a group with 2006 elements.\\
\item Suppose $f: G \rightarrow A$ is a group homomorphism, $A$ is abelian. Prove any subgroup of $G$ which contains $\operatorname{ker} f$ is normal. \\
\item If $G$ is a group, let $D = \{(x,x): x \in G\} \le G \times
G$. Prove that $G$ is a simple group if and only if $D$ is a
maximal subgroup of $G \times G$.\\
\item Let $P$ be a finite $p$-group. Prove the center of $P$ is nontrivial.\\
\item Recall that for subgroup $H \leq G$ we have the \emph{normalizer} of $H$:
$$N_G(H)=\{g \in G \mid gHg^{-1}=H\}$$ \noindent and the \emph{centralizer} of $H$:
$$C_G(H)=\{g \in G \mid gh=hg \,\,\forall h \in H\}.$$
\noindent Let $G$ be a finite group and $P$ a Sylow $p$-subgroup of $G$. Let $\pi$ be the permutation representation of $G$ acting on the left cosets of $N_G(P)$. Prove:\\
(a) $\pi(P)$ fixes exactly one letter (i.e. one coset).
(b) Suppose $|P|=p$ and let $x \in P$, $x \neq e$. Then $\pi(x)$
is a product of one 1-cycle and a certain number of $p$-cycles.
(c) If $|P|=p$ and $y \in N_G(P)-C_G(P)$, then $\pi(y)$ fixes at most $r$-letters where $r$ denotes the number of orbits under the action of $\pi(P)$.\\
\pagebreak
\section{Section II}
\item Let $I$ and $J$ be ideals in a commutative ring $R$ (with 1)
and suppose that $I + J = R$.
(a) Prove that $IJ = I \cap J$.
(b) Show that, as $R$-modules, $I \oplus J \cong R \oplus IJ$.
(c) Give an example of two such ideals $I$ and $J$ such that
neither is principal. [{\em Hint:} Consider $R = \mathbb{Z}[x]$.]\\
\item Let $F$ be a field, $F[x]$ and $F[x,y]$ polynomial rings in one and two commuting variables.
a. Prove $F[x]$ is a principal ideal domain. Determine all the maximal ideals.
b. Determine all the maximal ideals in $F[x,y]$. Is $F[x,y]$ a principal ideal domain? Explain.\\
\item Let $R$ be a commutative ring with identity. Prove that the subset of $R$ containing $0$ together with all zero divisors in $R$ must contain at least one prime ideal.\\
\item Let $R$ be a commutative Noetherian ring with 1.
(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 $R$ is not
Noetherian.\\
\item Let $\epsilon$ be a primitive $n^{th}$ root of unity in the
complex numbers. If $m$ is an integer such that $m > 2$, show that
the polynomial $x^m - 2$ has no roots in $\mathbb{Q}(\epsilon)$.
\pagebreak
\section{Section III}
\item Let $f(x) \in \mathbb{Q}[x]$ with deg $f$ = $n$ and let $K$
be a splitting field of $f(x)$ over $\mathbb{Q}$. Suppose that the
Galois group $G(K/\mathbb{Q})$ is isomorphic to the symmetric
group $S_n$.
(a) Show that $f(x)$ is irreducible over $\mathbb{Q}$.
(b) If $n > 2$ and $\alpha$ is a root of $f(x)$ in $K$, show that
the only automorphism of $\mathbb{Q}(\alpha)$ is the identity.
(c) If $n \ge 4$, show that $\alpha^n \not\in \mathbb{Q}$.\\
\item Prove the multiplicative group of nonzero elements in a finite field is cyclic.\\
\item
a. Write down a matrix which has characteristic polynomial
$c(x)=(x-1)^3(x-2)^3$ and minimal polynomial $m(x)=(x-1)^2(x-2).$\\
b. Are the two matrices below similar? Justify your answer:
$$A=\left(%
\begin{array}{rrr}
0 & 2 & 1 \\
-1 & 3 & 1 \\
1 & -1 & 1 \\
\end{array}%
\right), \,\, \left(%
\begin{array}{rrr}
0 & 2 & 1 \\
-2 & 5 & 2 \\
2 & -4 & -1 \\
\end{array}%
\right)$$\\
\item Let $A, B \in M_{n \times n}(\mathbb{C})$ such that $B$ is invertible. Prove there exists a scalar $\alpha \in \mathbb{C}$ such that $A+\alpha B$ is not invertible. \\
\item{True or false} All questions are for $n \times n$ matrices over $\mathbb{C}$ unless specifically stated.\\
a. The Jordan canonical form of a diagonal matrix is the matrix itself.\\
b. Matrices with the same characteristic polynomial are similar.\\
c. Every matrix is similar to its Jordan canonical form.\\
d. If a linear operator has a Jordan canonical form, then there is a unique Jordan canonical basis for that operator.\\
e. If the characteristic polynomial of $A$ has no multiple roots then $A$ is diagonalizable.\\
f. A matrix satisfying $A^2=A$ must be diagonalizable.\\
g. An invertible matrix $A$ is diagonalizable if and only if $A^{-1}$ is.\\
h. Interchanging two columns of a matrix preserves the determinant.\\
i. There exists a $5 \times 4$ matrix $A$ such that $AA^\tau$ is invertible.\\
j. The product of two eigenvalues of $A$ is also an eigenvalue of $A$.
\end{enumerate}
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