PH. D. COMPREHENSIVE EXAM — ALGEBRA — AUGUST, 1996

Notation: Z_{n} denotes the cyclic group of integers modulo n. F_{p} denotes the finite field with p elements. The symbols Q, R, and C denote, respectively, the fields of rational numbers, real numbers, and complex numbers. All rings are assumed to have an identity element. Grading: Four problems done completely (i. e., full credit on four problems) will guarantee a pass. It is probably wise to attempt more than four, in case you receive only partial credit on some of them.

1. (a) Find all subgroups H of order 4 of the group Z_{8} × Z_{2}.

(b) For each subgroup H from part (a), express (Z_{8} × Z_{2})/H as a direct product of cyclic groups of prime power order.

2. Let G be a group of order 255 = 3 · 5 · 17.

(a) Show that G has a unique 17-Sylow subgroup.

(b) Show that the 17-Sylow subgroup is contained in the center of G.

(c) Show that G is abelian.

3. Let R be a commutative ring and let

0 → M_{1} → M_{2} → M_{3} → 0

be an exact sequence of R-modules. Prove that M_{2} is Noetherian if and only if M_{1} and M_{3} are Noetherian.

4. Let R be a ring, I a minimal left ideal of R (i. e., the only left ideals that I contains are (0) and itself). Suppose I^{2} ≠ (0). Show that I = Re for some e ∈ I with e^{2} = e. (Hint: First show I = Ia for some a ∈ I. Then there exists e ∈ I such that ea = a. Show that this e is the desired element.)

5. Let K be the splitting field of x^{6} + 3 over Q.

(a) Compute the Galois group of K over Q.

(b) Find all fields intermediate between Q and K, indicating which are Galois over Q and which are not.

6. Let F be the splitting field of (x^{2} + 1)(x^{3} − x − 1) over the field K. Find the Galois group of F over K in each of the following cases:

(a) K = Q (b) K = Q(\sqrt{−23}) (c) K = R

(d) K = C (e) K = F_{5}