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\prof{Peter Cholak and Juan Migliore} \class{Math 222}
\quizdate{Monday, March 19, 2001} \pagestyle{empty}

\begin{document}

\noindent Quiz 4

\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper.\\

\begin{problem}
Find the following examples.  You do not have to prove your answers.
\begin{itemize}
\item[1.] A ring that is not commutative.

\item[2.] A commutative ring that is not unital.

\item[3.] A commutative unital ring that is not an integral domain.

\item[4.] An integral domain that is not a field.

\item[5.] A field that contains the rational numbers ${\mathbb Q}$ and
contains $\sqrt{3}$ but is not the whole set of real numbers.  
\end{itemize}
\end{problem}

\bigskip

\begin{problem} 
  Let $E$ be $\{a+b\sqrt{2}i\mid a,b\in\mathbb{Q}\}$, where $i$ is the
  complex number with $i^2=-1$.  We know that $E$ is a ring with usual
  addition and multiplication.
\par (1) Show that $E$ is a unital ring.\par (2) For any nonzero element
$x=a+b\sqrt{2}i\in E$, show that $x^{-1}\in E$.\par (3) Show that $E$
is a field.
\end{problem}

\bigskip

\begin{problem}
  Consider the set $\{0,2,4\}$ under the usual addition and
  multiplication modulo 6.  This is a ring (you don't have to check
that).  Is it a commutative ring? a
  unital ring?  a field? (Hint: write out a times table)
\end{problem}

\bigskip

\begin{problem}
  Let $p(x) = 2x^2 + 3x + 4$ and $q(x) = 4x^5 + 4x^4 + 4$.  Working
  in $\mathbb{Z}_5[x]$ find $d(x)$ and $r(x)$ such that $q(x) =
  d(x)p(x) + r(x)$ and the degree of $r(x)$ is less than the degree of
  $p(x)$.
\end{problem}


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