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\begin{center}
{\bf Test 2 for Honors Algebra III, Math 361 \vspace{3mm}\\ }
 {\bf Name:} {\footnotesize \hfill November 23, 1998}\\
\end{center}
Instructions: The test is 50 minutes in length.




\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Prove that there is no permutation $\sigma$ such
  that $\sigma(1,2,3)\sigma^{-1}=(1,2)$.

 
  \vfill (10 pts) Define the alternating group $A_n$ and find its
  cardinality.
  
  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Let $f(x)=x^4+3x^3+6x^2+3x+5\in\Q(x)$ and
  $g(x)=x^4+x^3+11x^2+x+10\in\Q(x).$ Find the greatest common
  divisor of $f(x)$ and $g(x)$.

 
  \vfill (10 pts) Show that $x^4+x^3+x^2+x+1 \in\Q(x)$ is
  irreducible.
  
  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Let $\Z_2$ be the field of two elements and let
  $R=\Z_2[x]$ be the polynomial ring. Prove that $f(x)=x^3+x^2+1$
  is an irreducible polynomial.
  
  
  \vfill (10 pts) What can you say about the ideal $(f(x))$. Give
  details on the structure of the quotient ring $R/(f(x))$ such
  as: How many elements does it have, what elements have a
  multiplicative inverse and are there any zero divisors?

  
  \vfill

  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (20 pts) Let $p$ be a prime. Show that every group of order
  $p^2$ is abelian.


  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (20 pts) Let $R=C[0,1]$ be the ring of real valued
  continuous functions on the interval $[0,1]$. (You can assume
  that $R$ is a ring). Show that
  $$M=\left\{ f\in R\mid f\left( \frac{1}{2}\right) =0\right\} $$
  is a maximal ideal. What can you say about $R/M$?

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{enumerate}
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