Masters Algebra Examination Oklahoma State University Department of Mathematics June, 1998

1. Exactly two of the following three groups are isomorphic.

ℤ/18ℤ × ℤ/4ℤ ℤ/36ℤ × ℤ/2ℤ ℤ/9ℤ × ℤ/8ℤ

Identify which two are isomorphic and prove that those two are isomorphic, and prove that they are not isomorphic to the remaining group.

2. (a) Define Sylow subgroups of a finite group, and state the Sylow theorems as completely as possible.

(b) Using these theorems, prove that there are no simple groups of order 56.

3. Let V be a vector space over a field F and let W ⊂ V be a subspace. Show that

dim(V) = dim(W) + dim(V/W).

4. (a) Show that ℚ[x] is a principal ideal domain.

(b) Let I be the ideal of ℚ[x] generated by the polynomials q_{1}(x) = 4x^{3} − 8x^{2} + 4x and q_{2}(x) = 2x^{3} − 6x^{2} + 4x. Find a polynomial p(x) such that I = (p(x)).

(c) Show that ℚ[x]/I is not an integral domain. Can I be a prime ideal?

5. Let f(x) ∈ ℚ[x] be a polynomial of degree 3, not necessarily irreducible.

(a) What are the possible Galois groups of f(x) over ℚ?

(b) Give examples of cubic polynomials f(x) with each possible Galois group from (a).

(c) For each polynomial in (b), give the splitting field E/ℚ of f(x) and the action of the Galois group on E as field automorphisms.