**Algebra Preliminary Exam, Spring 1986**

- 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*. - Let
*G*be an abelian group of order*p*^{6}, and let . Suppose that |*H*|=*p*^{2}. Give all possible such groups*G*. - Prove that
*S*_{4}is solvable. - In
*S*_{5}, how many Sylow subgroups of each type are there? -
- Let
*R*be a PID and let*S*be a multiplicatively closed subset. Show that*S*^{-1}*R*is a PID. - Give an example of a PID with exactly 3 nonassociate irreducible elements.

- Let
- 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 . - Let be a root of
*x*^{3}+4*x*+2.- Find a basis for over . Justify y our answer.
- Express in terms of the basis.
- Express in terms of the basis.

- Let
*K*be a subfield of the field*L*, and let such that is odd. Prove that .

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Wed Jul 31 14:18:12 EDT 1996