**Algebra Prelim, Fall 1990**

Answer all questions

- Give a complete list of all non-isomorphic abelian groups of order .
- Show that a group of order cannot be simple.
- Let
*G*be a finite*p*-group with |*G*| >*p*^{2}. Prove that*G*contains a normal abelian subgroup of order*p*^{2}. -
- Show that if
*G*is a subgroup of*S*_{n}, then either or . - Show that if and
*G*is a normal subgroup of*S*_{n}, then*G*=1,*A*_{n}or*S*_{n}. - Show that if , then
*S*_{n}has no subgroups of index 3.

- Show that if
- Let
*G*be an abelian group with 54 elements. Suppose that*G*cannot be generated by one element, but can be generated by two elements. Prove that*G*is isomorphic to . - Let
*K*be an extension field of*F*with [*K*:*F*] = 14. Let be a polynomial of degree 5. Suppose*f*(*x*) has no roots in*F*but has a root in*K*. What can you say about the factorization of*f*(*x*) into irreducibles in*F*[*x*] and*K*[*x*]? - Let
*f*(*x*) be irreducible over with splitting field*E*, and let and be roots of*f*in*E*. If has an abelian Galois group, prove that . - Let
*R*be a commutative ring with identity, and let be the set of maximal ideals of*R*. Let*A*be an ideal of the polynomial ring*R*[*x*] such that . Show that for some . (Hint: consider the set is a coefficient of some polynomial in*A*}.)

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Wed Jul 31 08:48:21 EDT 1996