Algebra Prelim, Fall 1990

Answer all questions

1. Give a complete list of all non-isomorphic abelian groups of order .
2. Show that a group of order cannot be simple.
3. Let G be a finite p-group with |G| > p². Prove that G contains a normal abelian subgroup of order p².
4. 1. Show that if G is a subgroup of S_n, then either or .
   2. Show that if and G is a normal subgroup of S_n, then G=1, A_n or S_n.
   3. Show that if , then S_n has no subgroups of index 3.
5. 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 .
6. 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]?
7. 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 .
8. 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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