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\begin{document}

\begin{center}
{\bf\large  ALGEBRA TEST \#3}
\end{center}

\vspace{1in}

This exam is due Monday, December 17, at 1pm.
You may use your book, but you may not
communicate with others concerning the exam.
In order to receive full credit your answer must be 
{\bf complete}, {\bf legible} and {\bf correct}. 

\vfill

\begin{center}
I have neither given nor received aid on this exam.
\end{center}

\vspace{.5in}

\begin{flushright}
{\bf Name:\underline{\hspace{2.5in}}}
\end{flushright}

\break

1. Let $\displaystyle \omega = e^{2\pi i/3}$
be a primitive third root of unity.
Show that ${\BZ}[\omega]$ is a Euclidean Domain.
(Hint: 
%elements of ${\BZ}[\omega]$ have the form
%$a + b\omega + c\omega^2$ where $a, b, c\in {\BZ}$.
If $\alpha = a+b\omega + c\omega^2$, $a, b, c\in {\BZ}$,
then define $N(\alpha) = \alpha\overline{\alpha} = 
(a+b\omega + c\omega^2)(a+b\omega^2 + c\omega) = 
((a-b)^2 + (b-c)^2 + (c-a)^2)/2$.)

\vfill

2. Find the integer solutions to $x^2 + y^2 = z^3$
that satisfy $\gcd(x,y)=1$.

\vfill

3. Let ${\BF}$ be a field. Show that ${\BF}[x]$
has an infinite set of pairwise nonassociate primes.

\vfill

4. Establish the irreducibility of $r$ in the integral
domain $\m d$:

\begin{enumerate}
\item[(a)] $r = x^n + px + p^2$, $\m d = {\BZ}[x]$, $p$ is
a prime integer.
\item[(b)] $r = y^2-x^3$, $\m d = {\BF}[x,y]$, ${\BF}$ is
a field and $n$ is an integer greater than $1$.
\end{enumerate}
\vfill

\end{document}
