Algebra Examination (Ph.D.)

Oklahoma State University, Department of Mathematics August, 1992

1. Let p and q be prime numbers with q < p.

(a) Prove that any finite group of order pq is solvable. (Use of the Sylow theorems is permitted.)

(b) Prove that the symmetric group S_{p} of degree p is generated by any 2-cycle and any p-cycle.

(c) Give a counterexample to part (b) if p is not prime.

2. Let R be a ring, and let N_{1} and N_{2} be submodules of the left R-module M such that N_{1} + N_{2} = M. Prove that M/(N_{1} ∩ N_{2}) is isomorphic to (M/N_{1}) ⊕ (M/N_{2}) as left R-modules.

3. Let A be a commutative ring with identity, and let 𝔞 be a proper ideal of A. (a) Prove that 𝔞 is contained in a maximal ideal 𝔪 of A.

(b) Let S be a multiplicative subset of A such that S ∩ 𝔞 = ∅. Prove that there is an ideal 𝔭 which is maximal subject to the conditions that 𝔞 ⊂ 𝔭 and 𝔭 ∩ S = ∅. Prove that 𝔭 is a prime ideal of A.

(c) Let 𝔞 = ⋂_{𝔭} 𝔭 be the intersection of all prime ideals of A. Prove that 𝔞 consists precisely of all elements x of A such that x^{n} = 0 for some integer n > 0.

(d) Suppose now that the ring A is Noetherian as well. Prove that the ideal 𝔞 considered in part (c) is nilpotent.

Please Turn Page 4. Let f(x) = x^{4} − 7.

(a) Determine the degree of the splitting field K of f(x) over ℚ.

(b) Determine all the subextensions F such that ℚ ⊂ F ⊂ K, and arrange these in

a graph ordered by inclusion.

5. (a) Define what it means for an irreducible polynomial f(x) ∈ ℤ[x] to be solvable

by radicals over ℚ.

(b) Prove that x^{5} − 9x + 3 is irreducible.

(c) Prove that x^{5} − 9x + 3 is not solvable by radicals over ℚ.