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\centerline{Mathematics 351 -- Abstract Algebra I}
\centerline{Midterm Exam 1 -- October 1, 2007}
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{\it Directions:}  Place all work you want to have considered for credit
in the blue exam book.  You have until 9:00pm to work on the exam if
you need that much time.  There are 100 possible points.
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I.  {\it Terminology.}
\item{A)} (5) Give an example of a ring $R$ that is {\it not} a {\it commutative ring}.
\item{B)} (5) What does it mean to say that $R$ is a {\it ring with identity}?
\item{C)} (5) What does it mean to say that $u \in R$ is a {\it unit}?
\item{D)} (5) Is every commutative ring with identity an {\it integral domain}?
If so, say why.  If not, give a counterexample.
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II.  Let $R$ be a ring.  Let $b$ be any one particular element of $R$, and
let
$$C_b = \{a \in R : a b = b a\}.$$
\item{A)}  (10)  Show that $C_b$ is a subring of $R$.  
\item{B)}  (10)  Let $S$ be a second ring, and let
$\varphi : R \to S$ be a ring homomorphism.  Show that if
$b \in R$, then $\varphi(C_b) \subseteq C_{\varphi(b)} \subseteq S$.  
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III.  Let $F[x]$ be the ring of polynomials with coefficients in a field
$F$.
\item{A)} (10) Show that the quotient and remainder under the division 
of $f(x)$ by $g(x)$ in $F[x]$ are unique.
\item{B)} (5)  {\it Define:}  $f(x)$ is {\it irreducible} in $F[x]$.
\item{C)} (10)  In what sense are factorizations of polynomials into 
irreducibles unique?  (State the theorem that answers this question; you
do not need to prove it.)
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IV. 
\item{A)} (10)  Let $F$ be a field.  
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) Let $F$ be a field.  If $a_1,\ldots,a_n$ are {\it distinct}
roots of $f(x) \in F[x]$ in $F$, show that 
$\left((x - a_1)(x - a_2)\cdots (x - a_n)\right)|f(x)$.
(Hint:  induction on $n$).  
\item{C)} (5)  What are the roots of $x^2 + 2x \in {\bf Z}_8[x]$ in ${\bf Z}_8$?  
Why is your result {\it not a contradiction} to part B?
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V.  All parts of this question refer to 
$$f(x) = 5x^5 - 12 x^3 + 36 x^2 + 18\in {\bf Q}[x].$$
\item{A)} (5) List all rational numbers that could be roots of $f$ according 
to the Rational Root Test.  
\item{B)} (5) Use any applicable method to decide whether $f$
is irreducible in ${\bf Q}[x]$, and state which criterion or criteria you are using.  
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