\documentclass[]{article}

\usepackage{epsfig}
\usepackage{latexsym}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{graphicx}
\pagestyle{empty}
%\setlength{\topmargin}{-1in}
%\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{8in}
%\setlength{\textwidth}{7.2in}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\mcg}{\vspace{0.3cm}}
\newcommand{\blank}{\underline{\hspace{2cm}}}
\newcommand{\solution}[1]{\parbox{1cm}{} \hfill {#1}\parbox{3cm}{: \hrulefill} \\}
\begin{document}

\begin{center}
\Large Prep Exercises for Exam \#2
\end{center}
{\Large M366} \hfill {\Large Spring 2012} \\



\begin{enumerate}

\item Define the $gcd$ of two positive integers. 

\item 
 \textsc{TRUE-FALSE:} 

\begin{enumerate}

\item Suppose that $gcd(a,b)=24$ Can $20$ be a divisor of $a$?


\item Suppose that $gcd(a,b)=24$ Can $20$ be a divisor of both $a$ and $b$?


\item Assume there exist two integers $r$ and $s$ such that
$$
ar +bs= 8.
$$
Can you conclude that $a$ and $b$ are both even?




\item Suppose you are given integers $a$ and $b$,such that for every pair of integers $r,s$, we have that
$$
ar +bs > 0
$$
implies
$$
ar +bs\geq 8.
$$
 Further, there exist two integers $r$ and $s$ such that
$$
ar +bs= 8.
$$
Then:
\begin{enumerate}
\item $a$ and $b$ are both even.
\item $a$ and $b$ are both divisible by $8$.
\item $a$ and $b$ can both be divisible by $16$.
\end{enumerate}



\item The $gcd$ of two odd numbers can be even.

\end{enumerate}

\item If $p$ is a prime number, what can the $gcd(a,p)$ be? 


\item Give an example of:
\begin{enumerate}
\item A field.
\item A commutative ring.
\item A non-commutative ring.
\item A ring homomorphism from $\mathbb{Z}/25\mathbb{Z}\to\mathbb{Z}/5\mathbb{Z}$.
\item A ring homomorphism from $\mathbb{Z}/5\mathbb{Z}\to\mathbb{Z}/25\mathbb{Z}$(What do you notice?).
\item  A  ring isomorphism.
\end{enumerate}

\item What is
$$ \sum_{k=0}^{n} {{n}\choose{k}} ?$$
Show that this computation gives yet another proof of the fact that the power set of a set $X$ with $n$ elements has cardinality $2^n$.

\item What is
$$ \sum_{k=0}^{n}(-1)^k {{n}\choose{k}} ?$$
\item What is
$$ \sum_{k_1+k_2+k_3=n} {{n}\choose{k_1,k_2,k_3}}? $$


\item 
Prove that for $n\geq 4$
$$
n! \geq 2^n 
$$

\item
Let $X$ be a set with $n$ elements and $a_1, \ldots, a_k$ be $k$ natural numbers  such that $\sum a_i=n$. Prove that the number of possible ways of partitioning $X$ into $k$ subsets $A_1, \ldots, A_k$ with $|A_i|=a_i$ is
$$
{{n}\choose{a_1, \ldots, a_k}}:=\frac{n!}{a_1!a_2!\ldots a_k!}
$$

%\pagebreak
%\item(15 pts.)\begin{enumerate} 
%\item Write the definition of a \textit{prime number}.
%\item Prove that there are infinitely many prime numbers.
%\end{enumerate}

\item Prove that $\mathbb{R}[x]$ is not a field.

\item Consider the ring $\bZ/12\bZ$. What elements have a multiplicative inverse? What elements are zero divisors? Can you make a statement for the general case of $\bZ/n\bZ$? (You may want to do more than one example before you tackle the general case!)

\item Write the definition of an ideal of a ring.

\item Prove that the set of all numbers that are divisible by 6 is an ideal in $\bZ$. What is the quotient ring isomorphic to? What is the image of the number 27 under the projection map?

\item Consider the ring $R=\mathbb{R}[x]$ and the subset $I$ of  polynomials such that $p(0)=0$.
\begin{enumerate}
	\item Is $I$ an ideal?
	\item What is the quotient ring $R/I$ isomorphic to?
	\item What element does the polynomial $x^7+ x^5-12$ correspond to under this natural isomorphism?
\end{enumerate}


\item Consider the ring $R=\mathbb{R}[x]$ and the subset $I=\{p(x) \ divisible\  by\  (x^2+5)\}$.
\begin{enumerate}
	\item Is $I$ an ideal?
	\item Describe the quotient ring $R/I$.
	\item What element does the polynomial $x^3$ correspond to under this natural isomorphism?
\end{enumerate}



\end{enumerate}	
\end{document}