\magnification=\magstephalf
{\bf \centerline{Math 361 Final Exam;  Tues Dec 16, 1997
8:00--10:00am}}\bigskip \noindent{\bf Instructions.} Answer questions
1--5.  You must show all
necessary working to receive full 
points for  a problem.\bigskip
\noindent{\bf 1.} {\sl (30 points)}\smallskip
\item{(a)} Find all the rational numbers which are roots
of the polynomial $X^4+2X^3+X+2$, and hence factor this polynomial into a
product of monic irreducible polynomials in ${\bold Q}[X]$.\smallskip
 \item{(b)} State Eisenstein's
criterion and use it to show that the polynomial $$f(X)=20 X^5-12  X^4 +15 X^3
-63 X^2 +30X-60$$ is irreducuble over the rational numbers.
\smallskip 
\item{(c)} Let $g=(1,2)(3,4,5,6)$ and $h=(6,5)(4,3,2,1)$ in the symmetric
group $S_6$ on six letters. Find an element $x\in S_6$ so $g=xhx^{-1}$.
\bigskip
 \noindent{\bf 2.} {\sl (30 points)} Let $G$ be a  group and $H$ be a subgroup
of $G$.\smallskip \item{(a)} Carefully explain what is meant by the left coset
$gH$ of an element $g$ of $G$.\smallskip \item{(b)} Define the index $[G\colon
H]$ of $H$ in $G$.\smallskip \item{(c)} Suppose that $G$ is finite. Define the
order $\vert G\vert$ of $G$. What is the relationship between the orders of
$G$ and $H$, and the index $[G\colon H]$?\smallskip
\item{(d)} Define the order of an element $g\in G$. Use your answer to
(c) to show that if $G$ is a finite group, the order of $g$ divides the order
of $G$.
\bigskip 

\noindent{\bf 3.}  {\sl(30 points)} In this question, $G$ denotes the subset
$G=\{ \pm 1, \pm i,\pm j,\pm k\}$ of $8$ elements of the quaternions. You may
ASSUME that  $G$ is a group under the usual multiplication of quaternions. 
\smallskip
\item{(a)} Show that $N=\{\pm 1\}$ is a normal subgroup of $G$.\smallskip 
\item{(b)} Explicitly list the elements of the cosets $gN$ of $N$ in $G$.
\smallskip
\item{(c)} Write down the multiplication table for the quotient group $G/N$.
\smallskip
\item{(d)} Show that $G/N$ is isomorphic  
 to ${\bold Z}_2\times {\bold Z}_2$ by explicitly describing an isomorphism
between these two groups.
\bigskip

\noindent{\bf 4.} {\sl (30 points)} Let $R$ be a ring, and $I$, $J$ be
two ideals of $R$.\smallskip
 \item{(a)} Define   the terms ring and ideal.\smallskip
  \item{(b)} Show carefully that $I\cap J$ is an
ideal of $R$, and that if $x\in I$ and $y\in J$, then $xy\in I\cap
J$.\smallskip \item{(c)} Explain what is meant by saying that $R$ is an
integral domain.\smallskip  \item{(d)}  Suppose  that $R$ is an integral
domain and
  $I\cap J=\{0\}$. Use your answer to (b) to show that either $I=\{0\}$ or
$J=\{0\}$. 
\bigskip 
\noindent{\bf 5.} {\sl (30 points)} Let $R={\bold Z}_2[X]$ denote the ring of
polynomials in the variable $X$ over the field of $2$ elements.
Set $a(X)=X^4+X+1\in R$ and $b(X)=X^2+1\in R$.\smallskip
 \item{(a)} Prove that $a(X)$ is an irreducible polynomial in $R$.
Is $b(X)$ irreducible?\smallskip
\item{(b)} Find explicitly  polynomials $f(X)$ and $g(X)$ so that $f(X)
b(X)+g(x)a(X)=1$.\smallskip
\item{(c)} Explicitly list the distinct  elements of the ring $R/I$
where $I=(a(X))$ is the principal ideal generated by $a(X)$. 
Explain why $R/I$ is a field.
How many elements does $R/I$ have?\smallskip 
\item{(d)} In $R/I$,
prove that $({b(X)+I})^2=X+I$, and calculate $(b(X)+I)^{-1}$ explicitly.  
\bigskip




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