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\centerline {\bf PhD Algebra Exam}
\centerline {\bf Fall 1989}
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\centerline {Part I:   Do three of these problems.}
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1.  Let $G$ be a group, $I(G)$ the set of inner automorphisms of $G$.
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	\item "a)"  Show $I(G)$ is a group.
	\item "b)"  Show  $G/Z(G)  \sim  I(G)$.   ($Z(G) = \text{ center of }G$)
 \endroster
 
2.  Let $R$ be a commutative ring, $I$ an ideal of $R$.  Define the annihilator
$A(I)$ by $A(I) = \{ x \in R : xI = 0 \}$. Show $A(I)$ is an ideal of $R$, and
in each of the following cases, find $A(I)$.
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	\item "a)"  $R = \Bbb Z_{60}$,  $I = (18)$
	\item "b)"  $R = \Bbb Q[x]/(x^2-4)$,  $I = (x^2-4x-4)$
	\item "c)"  $R = \Bbb Q[x]/(x^2-2x+1)$,  $I = (x^4-1)$
 \endroster

3.  Let $V$ be a finite dimensional vector space over any field, $W$ a subspace. 
If $n = \dim V$, $r = \dim W$, show that $\dim(V/W) = n-r$.

4.  Let $F$ be a finite integral domain.
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	\item "a)"  Show that $F$ is a field.
	\item "b)"  Show $|F| =  pr$  for some prime $p$.
	\item "c)"  Show by example that $r >1$ is possible.
 \endroster
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\centerline {Part II:   Do two of these problems.}
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5.  Let $G$ be a finite group such that  $|G|$ is the product of distinct
primes.  Suppose that for each prime dividing $|G|$ the corresponding Sylow
subgroup is unique.  Prove $G$ is Abelian, and thus cyclic.

6.  For each of the following field extensions, determine the group of
automorphisms (fixing the ground field) and state (and explain) whether or not
the extension is Galois.
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	\item "a)"  $\Bbb Q(\root 6 \of 2)/ \Bbb Q(\root 3 \of 2)$
	\item "b)"  $\Bbb Q(\root 6 \of 2)/ \Bbb Q(\sqrt 2)$
	\item "c)"  $\Bbb Z_p(x) / \Bbb Z_p(x^p)$,  $p$ a prime
	\item "d)"  $\Bbb Q(\root 4 \of 2, i) / \Bbb Q$
	\item "e)"  $\Bbb Q(z) / \Bbb Q$  where $z = \text{ primitive }15^{\roman{th}}$ root of
$1$
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7.  Let $R$ be the subring of $\Bbb Q(x)$ consisting of rational functions whose
denominators have non-zero constant term,  $R = \{f/g : g(0) \ne 0\}$.
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	\item "a)"  Find all the units of $R$
	\item "b)"  Find all the maximal ideals of $R$
	\item "c)"  Find all the ideals of $R$
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