PH. D. COMPREHENSIVE EXAM — ALGEBRA — JANUARY, 1997

Notation: The symbol Q denotes the field of rational numbers. All rings are assumed to have an identity element.

1. Let G be a group of order 175 = 5^{2} · 7. Show that the 7-Sylow subgroup is contained in the center of G.

2. Let K be a field and U a finite subgroup of the multiplicative group K^{×} =^{def} K − {0}. Prove that U is cyclic.

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 and let I ⊆ R be a two-sided ideal. Suppose there exists e ∈ I such that ea = ae = a for all a ∈ I.

(a) Prove that xe = ex for all x ∈ R.

(b) Show there exists a two-sided ideal J of R such that R = I ⊕ J.

5. Let F be the splitting field of x^{4} − 7 over Q.

(a) Find the Galois group of F over Q.

(b) Find all fields K, Q ⊆ K ⊆ F, that are normal over Q.

6. (a) Find the minimal polynomial of \sqrt{2} + \sqrt{3} over Q.

(b) Find the Galois group of the splitting field of the polynomial from part(a) over Q.