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\begin{document}

\begin{flushleft}
{\bf\large  ALGEBRA\\
MIDTERM EXAM}
\end{flushleft}

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


This exam is due Tuesday, October 1.
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}. 

\medskip

1. True or False. Give a reason or
to support your answer.
\begin{enumerate}
\item[(i)] The prime numbers are well-ordered
in their usual ordering.
\end{enumerate}

\vspace{1in}

\noindent
{\it Solution 1.}
Prime number are a subset of $\BZ$ using least
integer theorem, there exists $p\leq p_i$. Assume $p_k\leq p$
by least integer theorem, $p\leq p_k$ so $p = p_k$.
If $p\neq p_k$ there is a contradiction.

\vspace{1in}

\noindent
{\it Solution 2.}
\underline{T}; the prime numbers are a subset of 
$\BZ$. The Least Integer Theorem states that 
in such a subset, there must exist a least
member, $p$. $p\leq p_i$ for any $i\in {\BZ}$.
Assume $p_k\leq p$ for some integer $k$.
From the above theorem,, we know that $p\leq p_k$.
The only way for both of these cases to be true
is if $p=p_k$. If $p\neq p_k$ there is a contradiction.  


\end{document}

