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\centerline {\bf PhD Algebra Exam}
\centerline {\bf Spring 94}
\medskip
\centerline {Part I:   Do three of these problems.} 
\medskip


1.  A group G is called a $p$-group (for $p$ a prime) if the order of $G$ is a
power of $p$.
 \roster
	\item "a)"  Show that if $G$ is a $p$-group and $A$ is a normal subgroup $G$ of
order $p$, then $A$ is contained in the center of $G$.
	\item "a)"  Give examples to show that this does not hold if  $| A | = p^2$, or if $
A| = p$ and $G$ is not a $p$-group.
 \endroster
 
2.  Let $T$ be a linear transformation on a finite dimensional vector space $V$,
such that $T^2 = T$.
 \roster
	\item "a)"  Show that $V = T(V) \oplus \ker T$
	\item "b)"  What is the matrix of $T$ with respect to a basis chosen according
to this direct sum decompostion ({\it{i.e.}}, conjunction of basis of $T(V)$ with
basis of $\ker T$)?
	\item "c)"  Compute such a basis for  $T = \left[\matrix 3 & -6 \\ 1 &
-2\endmatrix\right]$  acting on $\Bbb R^2$.
 \endroster
 
3.  Recall the following definitions:
	For a group $G$, subgroup $H \leq G$, we say $H$ is a characteristic subgroup if
every automorphism of $G$ sends $H$ into itself.  We denote $H_1 = [H, H] =$
commutator subgroup of $H$, and define the ``lower central series'' $\{G_i\}$ of
$G$ by $G_i = [G_{i-1}, G_{i-1}]$.
 \roster
	\item "a)"  Show that if $H$ is characteristic in $G$ then $H$ is normal in
$G$.
	\item "b)"  Show by induction that the lower central series subgroups are all
characteristic in $G$.
	\item "c)"  Compute the lower central series for the two non-abelian groups of
order $8$.
 \endroster

4.  Identify the splitting field of the polynomial $f(x) = x^3 - 2$ over each of
the following fields:
	a)  $\Bbb Z_2$\qquad	b)  $\Bbb Z_3$\qquad	c)  $\Bbb Z_5$\qquad	d)  $\Bbb Z_7$   

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

5.  Let  $S$ and $T$ be linear transformations on a finite dimensional vector
space $V$.
	\roster
 \item "a)"  Suppose $v$ is an eigenvector for both $S$ and $T$.  Show $v$ is
also an eigenvector for $S+T$ and for $ST$.  What is the relationship between the
corresponding eigenvalues?
	\item "b)"  Suppose $\lambda$ is a non-zero eigenvalue of $AB$.  Show that
$\lambda$ is also an eigenvalue of $BA$.  What is the relationship between the
corresponding eigenvectors?
 \endroster
 
6. %\runinitem
 \roster
 \item "a)"  Show that  $\Bbb Z[i]$ is a Euclidean ring.
 \item "b)"  Show that  $\Bbb Z[\sqrt{-5}]$ is not a Euclidean ring.
 \endroster
 
7.  Let $\zeta$ be a primitive $16^{\roman th}$ root of unity (so $\zeta^{16} =
1$) over the rationals $\Bbb Q$.
  \roster
 \item "a)"  Find the irreducible polynomial for $\zeta$ over $\Bbb Q$.
	\item "b)"  Identify the Galois group of $\Bbb Q(\zeta)$ over $\Bbb Q$.
	\item "c)"  How many subfields does $\Bbb Q(\zeta)$ have which are quadratic
over $\Bbb Q$?
 \endroster
 
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