\documentclass[12pt]{amsart}

\input{../../TeX/macros}
\usepackage{fullpage}

\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\M}{M}
\DeclareMathOperator{\Tor}{Tor}

\begin{document}

\title[101 Final Exam]{Math 101: Algebra I \\ Final Exam}
\date{Tuesday, 22 November 2016}

\maketitle

\hspace{0.5in}

\begin{center}
Name \ \underline{\hspace{3in}}

\vspace{1in}

\begin{tabular}{c|c}
\textbf{Problem} & \textbf{Score} \\
\hline
\ & \hspace{1in} \\
1 \\
\ \\
\hline
\ \\
2 \\
\ \\
\hline
\ \\
3 \\
\ \\
\hline
\ \\
4 \\
\ \\
\hline
\ \\
5 \\
\ \\
\hline
\ \\
6 \\
\ \\
\end{tabular}

\vspace{1in}

Total \ \underline{\hspace{1in}}
\end{center}

\newpage

\begin{problab}{1}
Let $F$ be a field.  For a group $G$ written multiplicatively, recall that the group algebra $F[G]$ is the $F$-vector space with basis $G$ and multiplication induced by the group law in $G$, extended $F$-linearly.

Let $G=S_3$, let $\tau=(1\ 2)$, and let $\alpha=1+\tau \in F[G]$.  

\begin{enumalph}
\item The element $\alpha$ acts $F$-linearly by left multiplication on $F[G]$:
\begin{align*}
T : F[G] &\to F[G] \\
\beta &\mapsto \alpha\beta
\end{align*}  
Compute the matrix of $T$ with respect to a basis of elements of $G$.  

\vspace{3in}

\item Compute the minimal polynomial and characteristic polynomial of $T$.

\vspace{1.5in}

\item Let $B=F[G]$, let $I=\{\alpha\beta : \beta \in B\}$ be the right ideal of $B$ generated by $\alpha$.  Observe that $I$ and $B/I$ are $F$-vector spaces, and compute $\dim_F I$ and $\dim_F (B/I)$.
\end{enumalph}
\end{problab}

\newpage


\begin{problab}{2}
Let $n \in \Z_{\geq 1}$.  Let $A \in \GL_n(\C)$ have $n$ distinct eigenvalues $\lambda_1,\dots,\lambda_n$.  Let $V=\M_n(\C)$.  Find the eigenvalues of the $\C$-linear map
\begin{align*}
T : V &\to V \\
M &\mapsto AMA^{-1}.
\end{align*}
\end{problab}

\newpage

\begin{problab}{3}
Let $p$ be an odd prime, and let $G=\GL_2(\F_p)$.  

\begin{enumalph}
\item Prove that a $p$-Sylow subgroup of $G$ is cyclic, and exhibit a $p$-Sylow subgroup of $G$.

\vspace{2.5in}

\item Give two different reasons why every $p$-Sylow subgroup of $G$ is conjugate to the one given in (a), at least one of which implies that any two \emph{generators} of any two $p$-Sylow subgroups are conjugate.  

\vspace{2.5in}

\item Show that there are exactly $p+1$ distinct $p$-Sylow subgroups in $G$.
\end{enumalph}
\end{problab}

\newpage

\begin{problab}{4}
Let $k$ be a field and let $R=k[x,y]$ be the polynomial ring over $k$ in the variables $x,y$.
\begin{enumalph}
\item Show that the ideal $(x) \subseteq R$ generated by $x$ is a projective $R$-module.

\vspace{1.5in}

\item Show that the ideal $(x,y)$ generated by both $x,y$ is \emph{not} a projective $R$-module.  \emph{[Hint: Show that the surjective $R$-module homomorphism $\phi:R^2 \to (x,y)$ defined by $\phi(e_1)=x$ and $\phi(e_2)=y$ does not split.]}
\end{enumalph}
\end{problab}

\newpage

\begin{problab}{5}
Let $R=\Z[i]$ where $i^2=-1$.

\begin{enumalph}
\item Compute a generator of the ideal $(3+11i,1+3i) \subseteq R$.

\vspace{2in}

\item Let $M$ be the $R$-module generated by $x_1,x_2,x_3$ subject to the relations
\begin{align*}
(i+1)x_2 + (i-1)x_3 &= 0 \\
6x_1 + (3i-1)x_2 - (i+9)x_3 &= 0
\end{align*}
Compute the rank of $M$ and the invariant factors of the torsion submodule $\Tor(M)$.
\end{enumalph}
\end{problab}

\newpage

\begin{problab}{6}
Let $R$ be a commutative ring and let $M,N$ be $R$-modules.
\begin{enumalph}
\item State the universal property of $M \otimes_R N$.

\vspace{2.5in}

\item Suppose that $R$ is a domain with field of fractions $F$, and that $N \subseteq M$ is an $R$-submodule such that $M/N$ is a torsion $R$-module.  Show that the inclusion $N \hookrightarrow M$ induces an $F$-vector space isomorphism
\[ N \otimes_R F \xrightarrow{\sim} M \otimes_R F. \]
\end{enumalph}
\end{problab}

\end{document}

\newpage

\end{document}