Algebra Examination (MS and EdD)

Oklahoma State University, Department of Mathematics

June, 1992

1. From basic definitions, prove that a subgroup H of a group G is normal if and only if it is the kernel of a group homomorphism f : G → J for some group J.

2. (a) State the fundamental theorem on the structure of finite abelian groups. (b) Determine up to isomorphism all abelian groups of order 32.

3. Let S_{n} be the symmetric group of degree n acting on the symbols {1, 2, ⋯, n}. (a) Determine the number of 5-Sylow subgroups of S_{5}.

(b) Prove that the normalizer of a 5-Sylow subgroup of S_{5} has order 20.

(c) Prove that every subgroup of order 20 in S_{5} is the normalizer of some Sylow

5-subgroup of S_{5}, and deduce how many subgroups of order 20 there are in S_{5}.

4. Let V be a vector space over the field K, and let T be a linear transformation of V into itself. Prove that any set {x_{1}, …, x_{n}} of eigenvectors of T corresponding to distinct eigenvalues is linearly independent.

5. Let K[x_{1}, …, x_{n}] be the ring of polynomials in n variables over the field K. (a) Prove that for n = 1 the ring K[x_{1}] is a principal ideal domain.

(b) Is K[x_{1}, x_{2}] a principal ideal domain? Prove your answer.

6. Let R be a principal ideal domain. Let I_{1} ⫋ I_{2} ⫋ I_{3} ⫋ ⋯ be a strictly increasing sequence of ideals of R. Prove that this sequence is finite.

7. Find the minimal polynomial of \sqrt{2} + \sqrt{3} over ℚ and determine its Galois group.