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\begin{center}
{\bf Final for Honors Algebra III, Math 361 \vspace{3mm}\\ }
 {\bf Name:} {\footnotesize \hfill December 15, 1998}\\
\end{center}
Instructions: The test is 120 minutes in length.




\begin{enumerate}
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\item {\sl (30 points)} Consider the set $\Z^3$ of three vectors
  with integer coefficients. Let
  $$H:= {\rm rowsp}_\Z\left(
   \begin{array}{ccc}5 & 4 & 2\\ -6 & 11 & 6 \\ -13 & 0 & 13
  \end{array}\right).$$
(a) Show that $\Z^3$ is a group with respect to vector addition
and $H$ is a subgroup.\smallskip\\
(b) Consider the factor group $\Z^3/H$. Determine this factor
group as good as you can. Determine for example: Is the group
infinite or finite. If it is finite how many elements does it
have? Is the group Abelian or not? Is the group cyclic or not?

  
  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\sl (30 points)} Let $\Z_2$ be the field of two elements.
  $$G=\left\{ \vier{a}{b}{c}{d}\mid a,b,c,d\in\Z_2\mbox{ and
      }ad-bc=1\right\}.$$
  (a) Show that $G$ forms a group with
  respect to matrix
  multiplication.\smallskip\\
  (b) How many elements does $G$ have?\smallskip\\
  (c) Show that $G$ is isomorphic to a symmetric group by
  constructing an explicit isomorphism.

  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\sl (30 points)}\\ 
(a) Define the alternating group $A_n$. \smallskip\\
(b) If $n\geq 3$ show that every element is a product of
3-cycles.\smallskip\\
(c) Find a normal subgroup in $A_4$ of order 4.
  
  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\sl (30 points)} Let $R=\C[x]$ be the polynomial ring
  with coefficients in the complex numbers~$\C$.\smallskip\\
  (a) If $\varphi(x)\in\C[x]$ is a polynomial define what it
  means:  ``$z$ is a root of  $\varphi(x)$''.\smallskip\\
  (b) If $\varphi(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is a
  polynomial of degree $n$ show that $\varphi(x)$ can have at
  most $n$ roots in $\C$.\smallskip\\
  (c) Show that $\varphi(x)\in\C[x]$ has a double root at $z$ if
  and only if $\frac{d}{dx}(\varphi(z))=\varphi^\prime(z)=0.$\smallskip\\
  (d) Let
  $$M=\left\{ f\in \C[x]\mid f\left( 1\right) =0\right\}.
  $$
  Show that $M$ is a maximal ideal of $\C[x]$. What can you say about
  $\C[x]/M$?

  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\sl (30 points)} Let $R=\Z_2[x]$ denote the ring of
  polynomials in the variable $x$ over the field of $2$ elements.
  Set $f(x)=x^4+x+1\in R$ and $g(x)=x^2+1\in R$.\smallskip\\
  (a) Prove that $f(x)$ is an irreducible polynomial in $R$.
  Is $g(x)$ irreducible?\smallskip\\
  (b) Find explicitly polynomials $a(x)$ and $b(x)$ so that
  $a(x)f(x)+b(x)g(x)=1$.\smallskip\\
  (c) Explicitly list the distinct elements of the ring $R/I$
  where $I=(f(x))$ is the principal ideal generated by $f(x)$.
  Explain why $R/I$ is a field.
  How many elements does $R/I$ have?\smallskip \\
  (d) In $R/I$, prove that $({g(x)+I})^2=x+I$, and calculate
  $(g(x)+I)^{-1}$ explicitly.




 
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