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\centerline {\bf PhD Algebra Exam}
\centerline {\bf Spring 1989}
\medskip
\centerline {Part I:   Do three of these problems.} 
\medskip

 1.  Let $A$ be the real $3x3$ matrix all of whose entries are $1$;
 $$
 A = \left[ \matrix
 1&1&1\\
 1&1&1\\
 1&1&1
 \endmatrix\right]
 $$

	Find\roster
 \item "a)"  the eigenvalues of $A$
 \item "b)"  for each eigenvalue, a basis for the space of eigen vectors
 \item "c)"  the characteristic polynomial of $A$
 \item "d)"  the minimal polynomial of $A$
 \item "e)"  the Jordan normal form of $A$
 \endroster

 2.  Let $\Bbb Z_m$ and $\Bbb Z_n$  be the cyclic groups of orders $m$ and $n$.
 \roster
 \item "a)" Prove that $\Bbb Z_m \times  \Bbb Z_n$ is cyclic if and only if
$\roman{GCD}(m,n) = 1$.
 \item "b)"  Prove that every subgroup of a cyclic group is cyclic.
 \endroster

 3.  Let $R$ be an associative ring with identity such that every element is
idempotent; that is, $x^2 = x$ for all elements $x \in R$.  
 \roster
 \item "a)"  Prove that $R$ is commutative and has characteristic $2$.
 \item "b)"  Give two examples of such rings, one finite and one infinite.
 \endroster


 4.  True or false:  Justify if true, give counterexample if false.
 \roster
 \item "a)"  An algebraic extension of a field has finite degree.
 \item "b)"  A solvable group is abelian.
 \item "c)"  A unique factorization domain is a principal ideal domain.
 \item "d)"  An infinite field has characteristic zero.
 \item "e)"  If a group is abelian then every subgroup is normal.
 \endroster


 \head Part II :   Do two of these problems.\endhead

 5.  Let  $f(x)$ be an irreducible cubic polynomial over the rationals $\Bbb Q$
with at least one non-real root.  Let $\Bbb K$ be the splitting field of $f(x)$.
 \roster 
 \item "a)"  Show $[\Bbb K : \Bbb Q] = 6$
 \item "b)"  Show that the Galois group $G(\Bbb K/Q)$ is isomorphic to the
symmetric group $S_3$ .
 \item "c)"  Show that there exist irreducible cubics over $\Bbb Q$ whose Galois
groups are not isomorphic to $S_3$, and say what the group must be.
 \endroster

 6.  Let $A$ be an invertible matrix over a finite field $\Bbb F$.  
 \roster
 \item "a)"  Show that there is an integer $k$ such that $A^k = I$ (identity).
 \item "b)"  Suppose the characteristic of $\Bbb F$ is p, and let $a\ne 0$ be an
element of $\Bbb F$.  Find a value of $k$ which works for the matrix 
 $$
 A = \left[\matrix
 1&a\\
 0& 1
 \endmatrix\right]
 $$
 \item "c)"  Find a value of $k$ which works for the matrix
 $$
 A = \left[\matrix
 1&0\\
 0& a
 \endmatrix\right]
 $$
 \endroster
 7.  Let $p$ and $q$ be primes, not necessarily distinct.  Prove that any group
of order $p^2q$ is solvable; consider separately the cases $p = q$ and $p \ne
q$. (You may assume Sylow theory and the class equation.)
			
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