Qualifying Examination Algebra January 1996
- Let be a group epimorphism between
the finite abelian groups G and H. Suppose that G has order
and H has order .
What is the order of ?
Describe, up to isomorphism, the possible groups of that order.
- Suppose that G is a group of order p4 q5 where
p and q are distinct primes. Suppose further
that both a Sylow p-subgroup and a Sylow q-subgroup are
normal in G.
(i) Prove that , where A and B are subgroups
of orders p4 and q5 respectively.
(ii) Prove that G has a normal subgroup of order pq.
Prove that a group of order is not
Find the degree and a -basis of
over where is the rational numbers. Justify your
- Prove that in a principal ideal domain D, every
nonzero prime ideal is a maximal ideal.
Deduce that if K is an integral domain and
is a ring epimorphism
with , then K is a field.
Let R be a commutative ring. Prove that R has no nonzero nilpotent
elements if and only if
has no nonzero nilpotent elements for all prime
ideals of R (where denotes the
localization of R at the prime ideal ).
Is it true that R is a domain if and only if
is a domain for all prime ideals
- Suppose that M is an R-module with submodules A and
B such that . Prove that the submodule of M
generated by A and B is isomorphic to (the direct
sum of A and B).
- Suppose that the Galois group of a Galois extension
E over F is S6.
(i) Show that there are at least 35 proper
subfields between E and F.
(ii) Show that there is a subfield
L between E and F such that L is Galois over F, but
there is no subfield between E and L which is Galois over
(iii) What is the dimension of L over F?
Mon Jul 29 20:45:27 EDT 1996