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\centerline {\bf PhD Algebra Exam} \centerline {\bf Fall 1994} \medskip
\centerline {Part I:   Do three of these problems.}  \medskip


 1. Find all the integers which are orders of elements of the alternating group
$A_5$.  Count how many elements of $A_5$ are of each of those orders and describe
all the elements of each order.


2.  Give short explanations:
 \roster
 \item  Why is a field necessarily a p.~i.~d.?
 \item  Why is every subgroup of a solvable group solvable?
 \item  Why is a field extension of finite degree necessarily algebraic?
 \item  Why is the number of elements of a finite field necessarily a prime
power?
 \item  Why is an abelian simple group necessarily of prime order?
 \endroster

 3.  Suppose that $V$ is a finite dimensional vector space, $T$ a nilpotent
linear transformation on $V$.  Let $n = \dim V$.  Show that  $\dim T^k (V) \leq n
- k$,  and thus $T^n  = 0$.  What does it say about the Jordan normal form of
$T$ if all the inequalities are equality?  Give a non-trivial example of a $T$
for which some of the inequalities are not equality.

 4.  Consider the rings $A = \Bbb Q[x]/(x^2 - 2x)$, $B = \Bbb Q[x]/(x^2 - 1)$, 
$C = \Bbb Q[x]/(x^2)$.  ($\Bbb Q =$ rational numbers.)  Show that $A$ and $B$ are
isomorphic, but $B$ and $C$ are not isomorphic.

\medskip \centerline {Part II:   Do two of these problems.}  \medskip

 5.  Let $F$ be the finite field with $p$ elements ($p$ is a prime), and let
$GL(n, F)$ be the group of invertible $n\times n$ matrices with entries in $F$.
\roster \item Determine the order of $GL(2, F)$. \item Find a $p$-Sylow subgroup
of $GL(2, F)$. \item Determine the order of $GL(3, F)$. \endroster

 6.  Show that the only group of order $8$ which is isomorphic to a subgroup of
the symmetric group $S_4$ is the dihedral group $D_4$.


 7.  Determine the structure of $\Bbb Z_{15}^\times$, the group of units of $\Bbb
Z_{15}$ ($=$ integers mod $15$).  Let $K$ be the cyclotomic field $\Bbb Q(z)$,
where $z$ is a primitive $15^{\roman th}$ root of unity.  Exhibit an isomorphism
of the Galois group $G(K/\Bbb Q)$ with $\Bbb Z_{15}^\times$.

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