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\centerline {\bf PhD Algebra Exam}
\centerline {\bf Fall 1987}
\medskip
\centerline {Part I:   Do three of these problems.} 
\medskip
 
1. 	Let $G$ be a finite group.  
 \roster
 \item 		"a)"  Prove that if $\f(x) = x^2$ is a homomorphism $G \to G$ then $G$
is abelian.
 \item 		"b)"  Give an example of a finite non-abelian group $G$ such that 
$\psi(x) = x^4$ is a homomorphism.
 \endroster

2.	Let $A$ be a $3\times 3$ matrix with rational entries.  Recall that $A$ is
called nilpotent if for some positive integer $k$,  $A^k = 0$.
 \roster
 \item 		"a)"  Show that $A$ is nilpotent $\iff$ 0 is the unique eigenvalue of
$A$.
	\endroster
 \noindent Now let $A$ be nilpotent.
 \roster
 \item 		"b)"  Show that $A^3 = 0$.
 \item 		"c)"  Exhibit all possible Jordan forms of $A$.
 \endroster
 
3.	Let $R = M_2(\Bbb Q)$, the ring of $2 \times 2$ matrices over $\Bbb Q$.
 \roster
	\item 		"a)"  Prove that $R$ is simple; that is, $R$ has no proper non-zero
ideals.
	\item 		"b)"  Exhibit a proper non-zero right ideal of $R$.
 \endroster
 
4.	Let  $a = 1 + \sqrt 3$,  let  $\alpha = \sqrt a$, and let $K = \Bbb
Q(\alpha)$.
	\roster
 \item 		"a)"  Find the irreducible polynomial for $\alpha$ over $\Bbb Q$.
	\item 		"b)"  Let $F = \Bbb Q(a)$.  Show that $F$ is normal over $\Bbb Q$, and
$K$ is normal over $F$.
	\item 		"c)"  Show that $K$ is not normal over $\Bbb Q$.
 \endroster
 
 
\medskip
\centerline {Part II:   Do two of these problems.}
\medskip

5.	Prove that all groups of order $\leq$ 12 are solvable.


6.  	Recall that if $R$ is a ring with $1$, a unit of $R$ is an element with a
multiplicative inverse in $R$.   
	Consider $R = \Bbb Z [\sqrt q]$, for $q \in \Bbb Z $.   Define $N(a + b \sqrt
q) = a^2 - qb^2$.
	\roster
	\item 		"a)"  Show that $u \in R$ is a unit $\iff Nu = \pm 1$.
	\item 		"b)"  Find all the units of $\Bbb Z [\sqrt{-3}]$.
	\item 		"c)"  Show that $ \Bbb Z [\sqrt 2]$ has infinitely many units.
 \endroster

7.	Let $F = \Bbb Q(\root 4 \of 5)$.
	\roster
	\item 		"a)"  Find the degree and a basis of $F$ over $\Bbb Q$.
	\item 		"b)"  Find the group of automorphisms of $F$ over $\Bbb Q$ and describe
its  fixed field.
	\item 		"c)"  Describe the Galois closure of $F$ over $\Bbb Q$ and its Galois
group.
 \endroster
  
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