**Qualifying Examination Algebra January 1996**

*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.*G*is a group of order*p*^{4}*q*^{5}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*p*^{4}and*q*^{5}respectively.(ii) Prove that

*G*has a normal subgroup of order*pq*.*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.*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 of*R*?*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*).*E*over*F*is*S*_{6}.(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*L*.(iii) What is the dimension of

*L*over*F*?

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Mon Jul 29 20:45:27 EDT 1996