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\centerline{Mathematics 352 -- Abstract Algebra II}
\centerline{Midterm Exam 1 -- February 27, 2008}
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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 regular and 10 Extra Credit 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 let $x$ in $G$.
What is the {\it conjugacy class} of $x$ in $G$?
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\item{B)} (5) Let $K$ be an extension field of the field $F$ and 
let $u \in K$.  What does it mean to say that $u$ is {\it transcendental}
over $F$?  
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\item{C)} (5) Let $K$ be an extension field of the field $F$.  What
does it mean to say that $K$ is {\it normal} over $F$? 
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II. 
\item{A)} (10)  State the Structure Theorem for finite abelian groups.
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\item{B)} (10)  Let $G = {\bf Z}_{42} \oplus {\bf Z}_{36} \oplus {\bf Z}_{84}$.
What are the elementary divisors and invariant factors of $G$?  
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\item{C)} (5)  What is the Sylow $7$-subgroup of $G$ from part B?  Is this a cyclic group?  
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III.  
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\item{A)} (20) State and prove the First Sylow Theorem for finite groups.
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\item{B)} (10) Use the Sylow Theorems to show that every group of order 33
is cyclic.
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IV.  
\item{A)}  (5) Let $K$ be an extension field of $F$, and let $u\in K$.  Show that
if $u^2$ is algebraic over $F$, then $u$ is also algebraic over $F$.
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\item{B)}  (10)  Show that ${\bf Q}(\sqrt{2},\sqrt{3})$ is a splitting 
field of $f(x) = x^4 + 2x^3 - 8x^2 - 6x - 1$ over ${\bf Q}$.  (Hint: Look for
quadratic factors of $f(x)$.)
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V.  (15) Show that if $F \subseteq E \subseteq K$ are field extensions and 
$[E:F], [K:E]$ are both finite, then $[K:F] = [K:E][E:F]$.  
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{\it Extra Credit}  (10)  Let $p(x)$ and $q(x)$ be irreducible polynomials
in $F[x]$ such that $\deg p(x)$ and  $\deg q(x)$ are relatively prime integers.
Show that if $u$ is a root of $p(x)$ in
some extension field of $F$, then $q(x)$ is also irreducible in $F(u)[x]$.  

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