\documentclass{amsart}
\title{Problems for practice - Final}
\author{}

\usepackage[margin=1.25in]{geometry}
\usepackage{amsmath}
\usepackage{amssymb}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.} }

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\FF}[1]{\mathbb{F}_{#1}}
\newcommand{\Stab}[2]{\operatorname{Stab}_{#1}\left(#2 \right)}
\newcommand{\Autg}[1]{\operatorname{Aut}_{\scriptscriptstyle{\text{group}}}\left( #1\right)}
\newcommand{\inv}{\operatorname{Inv}}
\newcommand{\Rad}[1]{\operatorname{Rad}\left(#1 \right)}
\begin{document}
\maketitle


\section{Groups}

{\bf List of examples to recall:}
\begin{itemize}
\item $S_n$ = the symmetric group on $n$ letters.\\
\item $D_{2n}$ = the dihedral group of size $2n$.\\
\item $C_n$ = cyclic group of size $n$, i.e, $C_n = \langle \sigma | \sigma^n=e\rangle
\cong \Z/n\Z$.
\end{itemize}


\begin{enumerate}
\item Consider the group $S_n$. Let $X = \{1,2,\cdots,n\}$ be the set with $n$ elements,
on which $S_n$ acts naturally. Let $P$ be the set of two-element subsets of $X$.
Fix $p = \{1,2\}\in P$. Prove that $\Stab{S_n}{p} \cong S_{2}\times S_{n-2}$. 
Prove that the action of $S_n$ on $P$ (extended naturally from the $S_n$--action 
on $X$) is transitive (meaning: there is only one orbit).
Conclude
that $\displaystyle |P| = \left(\begin{array}{c} n \\ 2\end{array}\right)$.\\

\item For $m,n\in \Z_{\geq 2}$ such that $\operatorname{g.c.d.}(m,n) = 1$, prove
that we have an isomorphism of groups $C_{mn} \cong C_m\times C_n$. Prove or disprove:
there is another isomorphism of groups: $\Autg{C_{mn}} \cong 
\Autg{C_m}\times \Autg{C_n}$.\\

\item State the classification theorem of finite abelian groups.\\

\item Let $G$ be a simple group (meaning: it has no non-trivial, proper, normal subgroups).
If $G$ is abelian, then prove that there is some prime $p\in \Z_{\geq 2}$ such that
$G\cong C_p$. If $G$ is not abelian, prove that $[G; G] = G$. Remember: we do not consider
the trivial group $\{e\}$ as simple.\\

\item Prove that there is no simple group with $126$ elements.\\

\item Let $p\in \Z_{\geq 2}$ be a prime number. 
Give an example of a non--abelian group with $p^3$ elements.\\

\item Prove that $D_{2n}\cong C_n\rtimes_{\inv} C_2$. Here $\inv : C_2
\to \Autg{C_n}$ is the group homomorphism that sends the non-trivial element of $C_2$
to $g \mapsto g^{-1}$ for every $g\in C_n$.\\

\item Recall the presentation of the dihedral group:
\[
D_{2n} = \langle \sigma, \rho | \sigma^2=e=\rho^n \text{ and } \sigma\rho\sigma = \rho^{-1}\rangle.
\]
Let $X = \{1,2,\ldots,n\}$ and consider the following action of $D_{2n}$ on $X$:
\[
\rho(i) = i+1 \text{ for } 1\leq i\leq n-1. \text{ And } \rho(n) = 1.
\]
\[
\sigma(k) = n-k+1 \text{ for } 1\leq k\leq n.
\]
Prove that $\Stab{D_{2n}}{1} \cong C_2$. Identify the non--trivial of this subgroup
of $D_{2n}$.\\

\item Let $p\in \Z_{\geq 2}$ be a prime number. Consider the group $G = \operatorname{GL}_2(\FF{p})$
of $2\times 2$ invertible matrices with entries from $\FF{p}$.
What is the size (or order) of $G$? Give an example of a Sylow $p$--subgroup of $G$.
Is your example a normal subgroup in $G$?

\end{enumerate}

\section{Rings}

\begin{center}
{\bf All rings considered below are commutative and unital. All rings
homomorphisms are assumed to preserve identities.}
\end{center}

\begin{enumerate}


\item Prove that $\Z[x]$ is not a PID, but is a UFD.\\

\item Prove that $\Z[\sqrt{-5}]$ is not a UFD.\\

\item Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f)$ is a field
if, and only if $f(x)$ is a non--zero irreducible polynomial.\\

\item Assume that $R$ is a ring such that $|R|<\infty$. Prove that
every prime ideal in $R$ is maximal.\\

\item Compute the group of invertible elements (units) in $\Q[x]/(x^6)$.\\

\item Let $n\in \Z_{\geq 2}$ and assume that $n = p_1^{a_1}\cdots p_r^{a_r}$
is the prime factorization of $n$. Prove that we have a ring isomorphism:
\[
\Z/n\Z \cong \Z/p_1^{a_1}\Z \times \cdots \times \Z/p_r^{a_r}\Z
\]

\item Let $R = \Z/125\Z$. Prove that every element in $R$ is either a 
unit or nilpotent. Recall the general statement and how we proved it:
for a ring $A$, a maximal ideal $M\subsetneq A$, and $n\in \Z_{\geq 1}$,
every element of $R = A/M^n$ is either a unit or nilpotent.\\

\item Let $R$ be a domain and $P\subsetneq R$ be a prime ideal. Prove that
we have an injective ring homomorphism: $R_P \to F(R)$. Recall that
$R_P = (R\setminus P)^{-1}(R)$ and $F(R) = (R\setminus \{0\})^{-1}(R)$
are obtained by inverting elements of $R$ which are not in $P$, and
non--zero elements of $R$, respectively.\\

\item Prove that $\Z[\sqrt{-1}]$ is a Euclidean domain.\\

\item Prove that $\Z[\sqrt{-3}]$ is not a UFD, by demonstrating
that $(1+\sqrt{-3})(1-\sqrt{-3}) = 4 = 2\times 2$ contradicts the uniqueness
axiom of a unique factorization domain. Or, you can argue that $2$
is an irreducible element, and yet $(2)$ is not a prime ideal - a thing
that is known to be true for UFD's. Recall $Z[\sqrt{-3}] \subsetneq
\mathcal{O}(\sqrt{-3}) = \Z\left[\frac{1+\sqrt{-3}}{2}\right]$, the
latter is a Euclidean domain, with the norm borrowed from
complex numbers!\\

\item Prove that every Euclidean domain is a UFD.\\

\item State the Eisenstein criterion for checking irreducibility of a polynomial
in one variable, with coefficients from a UFD.\\

\item Let $f(x)\in k[x]$, where $k$ is a field. Assume that the degree of $f(x)$
is 2 or 3. Prove that $f(x)$ is irreducible if, and only if $f(\alpha)\neq 0$
for every $\alpha\in k$. You may assume that $f(x)$ is monic, if you so wish.\\

\item Let $p\in \Z_{\geq 2}$ be a prime number. Consider the polynomial
$f(x) = x^p - x \in \mathbb{F}_p[x]$. Prove that $\displaystyle
f(x) = \prod_{\alpha\in \mathbb{F}_p} (x-\alpha)$.\\

\item Prove that $f(x) = x^3+3x^2+x+1\in \Z[x]$ is irreducible. (Hint: a linear change
$x\to x-a$ can get rid of $3x^2$ term. Determine this $a\in \Z$ and rewrite the polynomial
in this new variable.) Which result states that the irreducibility property is
same for $\Z[x]$ and $\Q[x]$ (under certain obvious condition)?\\

\item Prove that $x^2+x+1\in \mathbb{F}_2[x]$ is irreducible.\\

\item Let $k$ be a field and let $k(x)$ be the field of rational functions of $x$
with coefficients from $k$.
\[
k(x) = \left\{ \frac{p(x)}{q(x)} \text{ such that } p(x),q(x)\in k[x] \text{ and }
q(x)\neq 0 \right\}
\]

Consider the following subring of $k(x)$:
\[
R= \left\{ \frac{p(x)}{q(x)} \in k(x) \text{ such that } q(0)\neq 0
\right\}
\]
Prove that $R$ is a local ring, by explicitly writing its unique maximal ideal.
This includes proving that the set that you have written, is indeed the uniqule maximal
ideal. For extra credit: what is the set of prime ideals in $R$? \\

\item Give an example of an ideal $I\subsetneq R$ of a ring $R$ such that
$\Rad{I} = I$ but $I$ is not a primary ideal. {\em Hint: it will happen even for $R = \Z$.}\\

\item Let $R = \Q[x,y,z]/(z^2 - xy)$ and let $P = (x,z) \subset R$. Prove
that $P$ is a prime ideal. Prove that $P^2$ is not primary.\\

\end{enumerate}
\end{document}
