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\centerline{Mathematics 351 -- Abstract Algebra I}
\centerline{Midterm Exam 2 -- November 14, 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.  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.  {\it Terminology.}
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\item{A)} (5) Let $G$ be a group and $n$ be an integer.
What does it mean to say that $|g| = n$ for $g \in G$?
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\item{B)} (5) Let $I$ be an ideal in a ring $R$.  What does it mean to 
say that $a \equiv b \bmod I$?
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\item{C)} (5) Give an example of an ideal in a ring $R$ that is prime
but not maximal.
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II. (10)  Let $R$ be the ring of functions $f : {\bf R} \to {\bf R}$
that have derivatives of all orders.  Show that 
$I = \{f \in R : f(0) = f'(0) = 0\}$ is an ideal in $R$. 
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III. All parts of this problem refer to $p(x) = x^2 + x$ 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 
$${\bf Z}_2[x]/(p(x)).$$
\item{C)} (20) State and prove the First Isomorphism Theorem for rings. 
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\item{D)} (10) Let ${\bf Z}_2 \times {\bf Z}_2 = \{(a,b) : a,b \in {\bf Z}_2\}$, 
a ring under the component-wise sum and product operations.
Show that $f : {\bf Z}_2[x] \to {\bf Z}_2\times {\bf Z}_2$ defined by 
$f(g(x)) = (g(0), g(1))$ is a ring homomorphism and determine $\ker(f)$.
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\item{E)} (5)  Deduce that ${\bf Z}_2[x]/(p(x))$ is isomorphic
as a ring to ${\bf Z}_2 \times {\bf Z}_2$.
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IV. (10) Let $G = \{4,8,12,16\} \subset {\bf Z}_{20}$.  Is $G$ a group under
{\it multiplication} mod 20?  If so, say why and determine if $G$ is cyclic.
If not, identify which of the group axioms fail.
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V.  
\item{A)} (10) State Lagrange's Theorem for groups.  
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\item{B)} (5) Let $G$ be a group of order 60.  Is it possible for $G$ 
to contain elements $a,b$ with $|a| = 3$ and $aba^{-1} = b^4$?  Why or why not?
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