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\title[Final, Math 71, Fall 2012]{Final}
\author[Final, Math 71, Fall 2012]{Math 71, Fall 2012}

\begin{document}

\hfill Name: \underline{\hspace{150pt}}

\vfill

\maketitle

\centerline{\bf Due: November 19, 9am, 2012}


\thispagestyle{empty}

\vfill

{\bf Instructions:} \\

Include this cover sheet, signed, and the problem sheet along with your solutions. \\

Complete this exam alone, without collaboration with others, and without the use of prohibited resources. You may use your notes, slides from class posted on the course website, and the sections from the book that we have covered:\\
\centerline{Sections 0.1--1.3 (skip 1.4) 1.5--4.5 (skip 4.6) 5.1--5.2 (skip 5.3--6.3) 7.1--7.4 (skip 7.5, 7.6) 8.1--9.4.}
Do not use the internet (outside of the course web page), or any other outside resources in completing the exam. You may consult with Prof.\ Daugherty on this exam, but do not discuss anything related to the exam with others until Prof.\ Daugherty has announced that all exams are all handed in. 

\vfill

By my signature, I state that I completed this exam following the guidelines outlined above, as guided by the Dartmouth Academic Honor Principle. Namely, these solutions represent my own work without the use of prohibited resources. 

\vfill



\noindent{\bf Signature:\underline{\hspace{250pt}}}
\vfill

\pagebreak


{\small \begin{tabular}{|r|@{\qquad\qquad\qquad}|c}
\hline~\\
1.a&5pts \\~\\\hline~\\1.b&8pts \\~\\\hline~\\1.c&10pts \\~\\\hline~\\1.d&7pts \\~\\\hline~\\1.e&10pts \\~\\\hline~\\1.f&5pts \\~\\\hline~\\2.a&10pts \\~\\\hline~\\2.b&5pts \\~\\\hline~\\2.c&10pts \\~\\\hline~\\2.d&5pts \\~\\\hline~\\2.e&10pts \\~\\\hline~\\3.a&5pts \\~\\\hline~\\3.b&5pts \\~\\\hline~\\3.c&5pts \\~\\\hline~\\3.d&5pts \\~\\\hline~\\3.e&10pts \\~\\\hline~\\4&15pts \\~\\\hline~\\5&20pts \\~\\\hline\end{tabular}}



\pagebreak
\def\rad{\mathrm{rad}\,}
\def\Jac{\mathrm{Jac}\,}
\begin{enumerate}
\item {}[45 pts] {\bf The big group theory problem. } Suppose that $G$ is a group of order $693 = 3^2\cdot 7 \cdot 11$ with a normal Sylow 3-subgroup $P$. \\
{\small \sl (Hint: Recall that you know special facts about groups of order $\alpha^2$ and $\alpha \beta$, where $\alpha \neq \beta$ are primes. Also, it will often be useful that you know the exact size of specific Sylow subgroups.)}
\begin{enumerate}
\item {}[5 pts] Show that $G$ has a normal Sylow 11-subgroup $Q$.
\item {}[8 pts] If $p \in P$, show that $Q \subseteq C_G(p)$. \\
{\sl\small(Hint+: consider commutators and use part (a).)}
\item {}[10 pts] If $R$ is a Sylow 7-subgroup of $G$ and $p \in P$, show that $R \subseteq C_G(p)$. \\
{\sl\small (Hint+: Note that $R$ acts on $P$ by conjugation, and go back to your group action roots. It may be useful to show that $\{ x \in P ~|~ R_x = R \} \leq P$.)}
\item {}[7 pts] Conclude that $P \subseteq Z(G)$.
\item {}[10 pts] Conclude that $G$ is abelian.
\item {}[5 pts] Describe the possible isomorphism types of $G$.
\end{enumerate}

\vfill

\item {}[40 pts] 
Let $R$ be a commutative ring with 1. In ring theory, the term \emph{radical} gets used to talk about sets of elements that are badly behaved for one reason or another, and are important to understanding the structure of rings. 
\begin{enumerate}
\item {}[10 pts] The radical of an ideal $I$ is 
$$\rad I = \{ r \in R ~|~ r^n \in I \text{ for some } n \in \ZZ_{>0} \}.$$
Show that $\rad I$ is an ideal containing $I$. (You may use the Binomial Theorem in problem 7.3.25.)
\item {}[5 pts] Give an example of a ring $R$ and ideal $I$ where $I \neq \rad I$. 
\item {}[10 pts] Note that $\rad 0 = \{ x \in R ~|~ \text{$x$ is nilpotent } \}$ (review problems 7.1.13\&14). We call this ideal the \emph{nilradical} of $R$, denoted $\fN(R)$. Show that 
$$(\rad I )/I = \fN(R/I).$$
Use this result to conclude that the only nilpotent element of $R/\fN(R)$ is 0 (that $\fN(R/\fN(R))$ is trivial).
\item {}[5 pts] We call an ideal \emph{radical} if $\rad I = I$. Prove that all prime ideals are radical.
\item {}[10 pts] Define the Jacobsen radical $\Jac(R)$ of a ring is  
$$\Jac(R) = \bigcap_{\text{ maximal ideals $M$} } M. $$
Show that $\Jac(R)$ is a proper ideal of $R$ which contains $\rad 0$. 

\end{enumerate}
Note: The Jacobsen radical is the analog to the subgroup $\Phi(G)$ of ``non-generators'' from the midterm (called the \emph{Frattini subgroup}); by almost exactly the same reasoning, $\Jac(R)$ also the property that if $I$ is a proper ideal of $R$, then the ideal generated by both $I$ and $\Jac(R)$ is also proper.
\vfill


\pagebreak


\def\lcm{\mathrm{lcm}}
\item {}[30 pts] Recall from 8.1.11 that a \emph{least common multiple} of two elements $a,b \in R$ is $\lcm(a,b)=c$ if 
\begin{enumerate}[(i)]
\item $a | c$, $b | c$, and 
\item if $f$ is such
that $a | f$ and $b | f$, then $c | f$.
\end{enumerate}
You showed that if a least common multiple of $a$ and $b$ exists, then it is a generator for the unique largest principal ideal contained in $(a) \cap (b)$. 

\medskip

\noindent Let $R$ be a commutative PID with 1. Suppose that $I = (a)$ and $J = (b)$ are proper nonzero ideals in $R$.
\begin{enumerate}
\item  {}[5 pts] Show that $I \cap J = (\lcm(a,b))$.
\item  {}[5 pts] Show by example that we can have $IJ \neq I \cap J$ in a PID.
\item {}[5 pts]  Show that if $\gcd(a, b) = 1$, then $I + J = R$.
\item {}[5 pts]  Show that if $\gcd(a, b) = 1$, then $IJ = I \cap J$.
\item {}[10 pts] Show that if $\gcd(a, b) = 1$, then $R/(IJ) \cong R/I \times R/J$ \\
(Where the direct product of two rings is the product of the underlying groups together with the multiplication $(r_1, s_1)(r_2, s_2) = (r_1r_2, s_1 s_2)$, making it into a ring, possibly with $1 = (1,1)$. You may assume that, just like in group theory, that if $f: A \to B$ and $g: A \to C$ are ring homomorphisms, then $f \times g: A \to B \times C$ is as well.)
\end{enumerate}
\vfill


\item {}[15 pts] Within the world of commutative rings with 1:
\begin{enumerate}[($*$)]
\item Define integral domains.
\item Define the four kinds of integral domains that interest us most, give a qualitative reason for why we're interested in that kind of domain, and order them by containment (``Every $A$ is a $B$'').
\item You've now defined five kinds of rings. Give five examples of rings which show why the five definitions are distinct. (``Look, $A$'s exists, because of some example. Also, every $A$ may be a $B$, but not every $B$ is an $A$ because of some example.'')
\end{enumerate}

\vfill


\item {}[20 pts] List 10 or more (up to 20) group or ring theoretic properties about the integers.\\
(Show me your diversity of knowledge about abstract algebra. What kind of structure can one impose on a set? What kinds of questions does one ask about a set once that structure is imposed? In addition to giving properties that $\ZZ$ has, you can also list properties that $\ZZ$  doesn't have, or properties that $\ZZ$ endows on other sets. Ten legitimately diverse facts, including a couple of non-trivial statements, will receive full credit; if in doubt, list more or ask.) 
\end{enumerate}

\vfill
\end{document}  