ALGEBRA PRELIMINARY EXAMINATION Fall 1999

Attempt all four parts. Justify your answers.

\underline{Part I.}

1. Suppose G is a group of order 20. Show that G is abelian if and only if G has a normal subgroup of order 4.

2. Show that if a finite group of order n has a simple subgroup of order > n/3 and a proper normal subgroup of order 3, then we must have n = 6.

\underline{Part II.}

1. Let R be a commutative ring with 1 ≠ 0. If R has only finitely many maximal ideals and exactly one unit, then show that R is a finite ring.

2. Let R be a commutative ring with 1 ≠ 0 and x is an indeterminate. Show that if R[[x]] is a PID, then R is a field.

\underline{Part III.}

1. Let R be a commutative ring with 1 ≠ 0 and let M be a unitary R-module. Show that Hom_{R}(R ⊕ R, M) is a projective R-module if and only if M is a projective R-module.

2. Suppose M is a finitely generated abelian group. Show that M ⨂_{ℤ} M = 0 if and only if M = 0. Is the equivalence true for abelian groups that are not finitely generated ?

\underline{Part IV.}

1. Let K ⊂ F ⊂ E be fields such that F is finite galois over K with the galois group A_{7}. Assume for some u ∈ E, K(u) is galois over K with the galois group ℤ_{4} × ℤ_{7}. Determine the group Aut (F (u)/F).

2. Let K ⊂ F be finite fields and suppose g ∈ K[x] is irreducible in K[x] of prime degree. Show that either g is irreducible in F [x] or all its roots are in F.