\documentclass[11pt]{article}
\usepackage{amsmath}
\pagestyle{empty}
\newcommand{\sig}{\sigma}
\newcommand{\ra}{\rightarrow}
\newcommand{\eps}{\epsilon}
\newcommand{\ah}{\alpha}
\newcommand{\lam}{\lambda}
\newcommand{\ol}{\overline}
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}
\newcommand{\hsone}{\hspace*{1cm}}
\newcommand{\hstwo}{\hspace*{2cm}}
\begin{document}
\begin{center} {\Large\bf Honors Algebra I\\ Assignment 9} \\ Due,
Friday, Nov 6 \end{center}
\begin{quote}
The voyage of discovery lies not in seeking new horizons, but
in seeking with new eyes. \\ -- Proust
\end{quote}
\ben
\item Let $R$ be a ring. Call $a\in R$ a {\em unit } if
$ab=1$ for some $b\in R$. Let $X$ be the set of units
Prove that $X$ forms a group under multiplication.
What is our standard notation for $X$ in the case where $R=Z_n$?
\item Recall $Z[\sqrt{2}]=\{a+b\sqrt{2}: a,b\in Z\}$. Find a unit $\ah= a+b\sqrt{2}\in Z[\sqrt{2}]$
which has $a\geq 10$. (One approach: Find some unit $\beta$ other than $\pm 1$.
Then, as the units form a group, any power $\beta^w$ is also a unit.)
\item Let $R$ be a ring of characteristic $3$.
\ben
\item Prove that the
map $\phi:R\ra R$ given by $\phi(x)=x^3$ is a ring homomorphism.
\item Assume further that $R$ is an Integral Domain. Now prove
that $\phi$ is injective.
\item Assume yet further that $R$ is finite. Now prove $\phi$
is an isomorphism.
\een
\item Give a natural set of representatives for $Z[i]/(3)$.
Give the addition and multiplication tables for $Z[i]/(3)$.
Give the multiplicative inverse (you can read it off your
table!) for each nonzero element of $Z[i]/(3)$.
\een
\begin{quote}
Fearing error and fearing truth are one and the same. Those
who fear making mistakes are incapable of discovery. When we
worry about making mistakes, the error within us becomes an
unmovable rock. In our fear, we cling to what we have declared
to be ``true" one day, or what has always been presented as such.
When we are driven by a thirst for knowledge, and not by the
fear of seeing a false security fade away, then error, like
suffering and sadness, passes through us without ever gaining
substance, and the trace it leaves is that of renewed knowledge.
\\ Alexander Grothendieck, {\em Rec\'oltes et Semailes}
\end{quote}
\end{document}