Qualifying Examination Algebra January 1996

1. 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.

2. 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.

3. Prove that a group of order is not simple.

4. Find the degree and a -basis of over where is the rational numbers. Justify your answer.

5. 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.

6. 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 of R?

7. 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).

8. 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 L.

   (iii) What is the dimension of L over F?

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