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\centerline{Mathematics 351 -- Abstract Algebra I}
\centerline{Makeup Midterm Exam 2 -- November 19, 2007}
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{\it Directions:}  Place all work you want to have considered for credit
in the blue exam book.  You have two hours to work on the exam if
you need that much time.  There are 100 possible points.  Some partial 
credit will be given for relevant definitions, statements of theorems, etc.  
Even if you cannot see how to complete some part, don't leave it blank.
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I. 
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\item{A)} (5) Let $G$ be a group and $H$ be a subgroup of $G$.
What is $|H|$?
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\item{B)} (5) Let $R$ be a ring, $I$ an ideal in $R$ and $a,b\in R$.
What is true about $a - b$ if the cosets $a + I$ and $b + I$ are equal?  
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\item{C)} (5) Are there any examples of ideals $I$ in rings that are 
maximal but not prime?  Why or why not?
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II. (10)  Let $R$ be the ring of $2 \times 2$ matrices with real entries.
Is
$$I = \left\{\pmatrix{0 & 0\cr 0 & r\cr} : r \in {\bf R}\right\}$$
an ideal in $R$?  Why or why not?
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III.  (20) State and prove the First Isomorphism Theorem for rings. 
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IV. All parts of this problem refer to $p(x) = x^2 + 1$ in ${\bf Z}_2[x]$.
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\item{A)} (5) How many distinct cosets of the ideal $(p(x))$ 
are there in ${\bf Z}_2[x]$?
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\item{B)} (10) Construct the coset addition and multiplication tables for the 
quotient ring ${\bf Z}_2[x]/(p(x))$.
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\item{C)} (10) Let $R = \{(a,b) : a,b \in {\bf Z}_2\}$, with operations $\oplus$ and
$\otimes$
defined by $(a,b) \oplus (c,d) = (a + c,b + d)$ and $(a,b)\otimes (c,d) = 
(ac, ad + bc)$.  You may assume without proof that
$R$ is a commutative ring with identity under these operations.  Show that
the mapping $f : {\bf Z}_2[x] \to R$ defined by $f(g(x)) = (g(1),g'(1))$ is a ring homomorphism,
where $g'$ denotes the derivative, computed with coefficients mod 2.
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\item{D)} (5)  Deduce that ${\bf Z}_2[x]/(p(x))$ is isomorphic
as a ring to $R$ from part C.
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V. (7) Let $G = \{6,12,18,24\} \subset {\bf Z}_{30}$.  Is $G$ a group under
{\it multiplication} mod 30?  If so, say why.
If not, identify which of the group axioms fail.
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VI.  
\item{A)} (10) State Lagrange's Theorem for groups.  
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\item{B)} (8) Show that if $G$ is a finite group of order $n$, then
$G$ has elements of order $k$ for every positive integer $k$ dividing $n$ 
{\it if and only if} $G$ is cyclic.  
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