Algebra Preliminary Exam, Spring 1986

1. Let F be the free group on the set , and let H be the normal subgroup of F generated by . Prove that F/H is isomorphic to the free abelian group on X.
2. Let G be an abelian group of order p⁶, and let . Suppose that |H|=p². Give all possible such groups G.
3. Prove that S₄ is solvable.
4. In S₅, how many Sylow subgroups of each type are there?
5. 1. Let R be a PID and let S be a multiplicatively closed subset. Show that S^-1R is a PID.
   2. Give an example of a PID with exactly 3 nonassociate irreducible elements.
6. Let be the set of maximal ideals in a commutative ring R with identity. Set . For , prove that if and only if 1+rs is a unit for all .
7. Let be a root of x³+4x+2.
   1. Find a basis for over . Justify y our answer.
   2. Express in terms of the basis.
   3. Express in terms of the basis.
8. Let K be a subfield of the field L, and let such that is odd. Prove that .

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