Algebra Prelim

June 3 1996

Provide proofs for all statements. If necessary, you may use the results from earlier parts of a problem, even though you may not have successfully proved them.

This is a three-hour examination. Do as many problems as you can. Remember, correct and complete solutions for fewer problems carefully presented carries much more weight than numerous incomplete attempts. Good luck!

Some useful notation: Z Z Z = The integers, Q Q Q = The rationals, R R R = The reals, C C C = The complex numbers, Z Z Z_{p} = The finite field with p elements.

1. (a) Describe the group of automorphisms of a cyclic group of order 11.

(b) Prove that every group of order 77 is cyclic.

2. If G_{1} and G_{2} are finite groups and p is a prime, prove that every Sylow p-subgroup of G_{1} × G_{2} is of the form P_{1} × P_{2} where P_{i} is a Sylow p-subgroup of G_{i}, i = 1, 2.

3. Let τ ∈ S_{n} be an n-cycle and let r be a positive integer. Let σ = τ^{r}. If σ(1) = 1, what is σ(2)? Why?

4. Let a, b ∈ R R R, R R R the field of real numbers. Let A = R R R[x]/(x^{2} + ax + b).

(a) Show that A is isomorphic (as a ring) to one of the following rings: C C C, R R R × R R R, R R R[x]/(x^{2}).

(b) Show that no two of C C C, R R R × R R R, R R R[x]/(x^{2}) are isomorphic as rings.

5. Let R = k[t^{2}, t^{3}], a subring of the polynomial ring k[t], k a field. Show that t^{2} is an irreducible element in R, but not a prime element in R. Is R a UFD? Why?

6. Consider the equation (7 + 4i)(7 − 4i) = (8 + i)(8 − i) in Z Z Z[i] where i = \sqrt{−1}. Can this equation be used to show Z Z Z[i] is not a UFD? If yes, why? If no, why not?

7. Let K be a field, and let f(x) = x^{9} − x ∈ K[x]. Let F be a splitting field of f(x) over K. Find [F : K] and Gal(f) when

(a) K = Q Q Q,

(b) K = Z Z Z_{3}.

8. Let σ, τ ∈ AutC C CC C C(x) where

σ(x) = ix, i = \sqrt{−1}, and τ(x) = \frac{1}{x} .

Let G = ⟨σ, τ⟩, the group generated by σ, τ.

(a) Find |G|.

(b) Let C C C(x)^{G} be the fixed field of C C C(x) under G. Find [C C C(x) : C C C(x)^{G}] and find C C C(x)^{G}.

9. (a) Give a vector space V and a linear transformation T : V ⟶ V such that T^{2}+7T +19Id_{V} = 0 (where Id_{V} : V ⟶ V is the identity transformation defined by Id_{V} (v) = v for all v ∈ V).

(b) The same problem as part (a) except that T : V ⟶ V such that T^{3} − 2T^{2} + 2T + Id_{V} = 0.