**
Qualifying Examination Algebra Fall 1995**

#1. Suppose that *G* is a group of order (35)^{3}. Show that
*G* has normal
Sylow 5- and 7-subgroups. Also show that *G* has a normal
subgroup of
order 25.

#2. Suppose that *G* is an abelian group isomorphic to
. Let *H* be a subgroup of *G* of order 27. Up
to isomorphism, describe *H*.

#3. Let be a group epimorphism with *G* finite,
let , and let *C* denote the centralizer of *f*(*g*) in *H*.

a). Prove that if *D* is a conjugacy class in *f*^{-1}(*C*), then
*f*(*D*) is a conjugacy class in *C*.

b). Prove that the order of the conjugacy class of *g* in
*f*^{-1}(*C*) is at most .

c). Prove that the order of the centralizer of *g* in *G* is
at least |*C*|.

#4. Suppose that *R* is a unique factorization domain which is
NOT a principal ideal domain.

a). Show that *R* must have at least two (nonassociate) prime
elements.

b). Show that *R* must have a nonprincipal maximal ideal.

#5. Let *R* be a ring with a 1 and suppose that *X* is an
*R*-module and *N* is a submodule
of an *R*-module *M*. Let denote the inclusion
map and let be the *R*-module homomorphism .
Prove that if

is a commutative diagram
for some *R*-homomorphism *f*, then *M* is isomorphic
to .

#6. Suppose that *E* is a Galois field extension of *F* with
[*E*:*F*]=*p*^{n} for some prime *p* and positive integer *n*. Show that
there are intermediate fields so that [*K*_{i}:*K*_{i-1}] = *p* and *K*_{i}
is Galois over *F* for *i* =1, ... , *n*.

#7. Prove that there is no finite field which is algebraically closed.

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Tue Jul 30 07:51:55 EDT 1996