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\centerline{Mathematics 351 -- Abstract Algebra I}
\centerline{Final Examination -- December 12, 2007}
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{\it Directions}
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Do all work in the blue exam booklet.  Do not place anything
you wish to have considered for credit on the exam sheet.
There are 200 points, distributed as indicated next
to the questions.
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{\it Notation:} In the following, unless specified otherwise, $F$ denotes
a field.
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I.  {\it Terminology}
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\item{A)} (5)  What does it mean to say that $R$ is
a {\it ring with identity}?
\item{B)} (5) What does it mean to say that $a \in F$
is a {\it root} of $f(x) \in F[x]$?
\item{C)} (5) What does it mean to say that $f(x) \in F[x]$
is {\it irreducible}?
\item{D)} (5) What does it mean to say that a subset $I$ of 
a ring $R$ is an {\it ideal} in $R$?
\item{E)} (5) What does it mean to say that a permutation
$\sigma \in S_n$ is {\it even}?
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II.  
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\item{A)} (15) State and prove the Eisenstein Irreducibility Criterion 
in ${\bf Q}[x]$.
\item{B)} (10) Use the Eisenstein criterion to show that
there are irreducible polynomials of every degree $\ge 1$
in ${\bf Q}[x]$.  
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III.  
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\item{A)} (10) Show that $a\in F$ is a root of 
$f(x) \in F[x]$ if and only if $(x - a)|f(x)$ in
$F[x]$.  
\item{B)}  (10) Express $f(x) = x^4 - 16$ as a 
product of irreducible polynomials in ${\bf Q}[x]$, in
${\bf R}[x]$, and finally in ${\bf C}[x]$.  
\item{C)}  (5) In what sense are your factorizations
from part B {\it unique}?
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IV.  Let $p(x) = x^2 + x + 1\in {\bf Z}_2[x]$.  
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\item{A)} (15)  Determine the addition and multiplication
tables for the quotient ring $R = {\bf Z}_2[x]/(p(x))$.
\item{B)}  (5) Is $R$ a field?  Why or why not?
\item{C)}  (5) Show that $x + 1\in{\bf Z}_2[x]/(p(x))$ 
is a root of the polynomial $y^2 + y + 1 \in {\bf Z}_2[y]$. 
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V.  
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\item{A)} (10) State the First Isomorphism Theorem for
rings.
\item{B)}  (10)  Let $\varphi$ be the mapping
$$\eqalign{\varphi : {\bf Z}[x] &\to {\bf Z}_3\cr
                          f(x) &\mapsto f(0) \bmod 3.\cr}$$
That is, $\varphi$ takes each polynomial
$$f(x) = a_nx^n + \cdots + a_1x + a_0$$ to the 
class of $a_0$ mod 3).  Show that $\varphi$ is a ring homomorphism
and show that its kernel is the ideal $(x,3)\subset {\bf Z}[x]$.
\item{C)} (5) Is the ideal $(x,3)$ in ${\bf Z}[x]$ maximal?
Why or why not?
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VI.  Let $R$ be a ring with identity.  
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\item{A)} (7.5) Show that the set of units in $R$ is a group
under the multiplication operation from $R$.
\item{B)} (10) Determine the multiplication table for the
 group of units in the ring
${\bf Z}_8 \times {\bf Z}_4 = \{(r,s) : r\in {\bf Z}_8
{\rm\ and\ } s \in {\bf Z}_4\}$ (componentwise
sum and product).    
\item{C)} (7.5) Is your group from part B {\it cyclic}? Why
or why not?
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VII.  Let $G$ be a group, and let $H$ be a subgroup of $G$.
Define 
$$N_H = \{x\in G: xHx^{-1} \subseteq H\},$$
called the {\it normalizer of $H$} in $G$.
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\item{A)} (10) Let $G = {\rm GL}(2,{\bf R})$ be the group 
of $2\times 2$ invertible matrices under matrix multiplication
and let
$$H =  \left\{\pmatrix{1 & r\cr 0 & 1\cr} : r \in {\bf R}\right\}.$$
Show that $H$ is a subgroup of $G$ and determine $N_H$ in this case.
\item{B)} (10) Show that for any subgroup of a general 
group, $H \subseteq G$, 
$N_H$ is a subgroup of $G$ containing $H$, and that $H$ is a normal
subgroup of $N_H$.
\item{C)} (5) Show that the quotient group $N_H/H$
for $G, H$ as in part A is isomorphic to the group 
${\bf R}^*\times {\bf R}^*$, where
${\bf R}^*$ denotes the multiplicative group 
of nonzero real numbers.  
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VIII.  
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\item{A)} (5) By Lagrange's theorem, what are the possible
orders of subgroups of $S_4$?
\item{B)} (10) Show that the subset $G \subset S_4$ defined by 
$$G = \{(),(1234),(13)(24),(1432),(24),(1234)(24),(13)(24)(24),(1432)(24)\}$$
is a subgroup of $S_4$.  Note that the second four elements are
equal to the compositions of the first four with the two-cycle $(24)$.
\item{C)} (10) Determine the left and right cosets of $G$
in $S_4$. Is $G$ a normal subgroup of $S_4$?  Why or why not?
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\centerline{\it Have a peaceful and joyous holiday season!}
\bye
