Master’s Comprehensive Exam—Algebra—August 1997

Notation. Let |G| denote the order of the group G. If η is a homomorphism and k ⩾ 1, η^{k} means the composition η ∘ ⋯ ∘ η of k copies of η. If R is a ring, R^{k} means the direct sum R ⊕ ⋯ ⊕ R of k copies of R.

1. Let G be a group. Let H and K be normal subgroups of G, with H ∩ K = {1}. (a) Prove that every element of H commutes with every element of K. Hint: for all h ∈ H and

k ∈ K, prove that hkh^{−1}k^{−1} = 1.

(b) Prove that the map H × K → G given by (h, k) ↦ hk is a homomorphism.

(c) Assume in addition that the groups are finite, and that |G| = |H|·|K|. Prove that H×K ≅ G.

(d) The map in (b) can fail to be a homomorphism if one or both of H, K is not normal. Let G = S_{3}. Give examples of subgroups H, K of S_{3}, at least one of which is not normal, and specific elements (h_{1}, k_{1}) and (h_{2}, k_{2}) ∈ H × K, which illustrate why the map can fail to be a homomorphism.

2. Let G be a group. Let H be a subgroup of G of index n. Prove that there is a normal subgroup N in G, with N ⊆ H, such that the index [G : N] is a divisor of n! . Hint: consider the action of G on the coset space G/H.

3. Let R be a ring with identity element 1. Let I and J be ideals of R.

(a) Define the ideal I + J.

(b) In the ring ℤ, what is (10) + (16)? Proof is not required, but explain briefly.

(c) Prove the following version of the Chinese Remainder Theorem. Assume I and J are relatively prime, meaning I + J = (1). Let a and b be any two elements of R. Prove that there is an x ∈ R such that x ≡ a (mod I) and x ≡ b (mod J).

4. Consider the free ℤ-module ℤ^{3}. Let K be the submodule generated by the 3-tuples (2, 4, 5) and (6, −4, 3). Let M = ℤ^{3}/K.

(a) Express M up to isomorphism as a direct sum of cyclic ℤ-modules.

(b) What is the torsion submodule of M?

(c) Does M have a submodule isomorphic to ℤ/(4)? Explain.

(d) Does M have a submodule M′ such that M/M′ is isomorphic to ℤ/(3)? Explain.

5. (a) Let n be a positive integer. Let F be a field of characteristic zero containing a primitive n-th root of unity ζ. Let a ∈ F, and let E be the splitting field of x^{n} − a over F. Prove that the Galois group of E/F is cyclic, and that its order is a divisor of n.

(b) Give an example where the order of the Galois group is strictly between 1 and n.

6. In this problem, ζ = ζ_{5} is a primitive fifth root of unity over ℚ. Let E = ℚ(ζ).

(a) Describe the Galois group G of E/ℚ. What is its order? Is it cyclic? (No proof required on part (a).)

(b) Choose a generator η of G. For the η you have chosen, make a table showing ζ^{i} and η(ζ^{i}) for i = 1, 2, 3, 4.

(c) Find an expression in ζ which is invariant under η^{2} but not invariant under η. Call this expression u.

(d) Let H be the subgroup of G generated by η^{2}. According to the Galois correspondence, there is a subfield K = Inv H ⊂ E consisting of the elements of E which are invariant under H. Prove that K = ℚ(u), showing both the (⊇) and (⊆) directions.

(e) Find the minimal polynomial of u over ℚ.

(f) Prove that K = ℚ(\sqrt{5}).

(g) Make a table showing all the subgroups of G, including G itself and {1}. List the subfield of E that corresponds to each subgroup under the Galois correspondence.

Typeset by AMS-TEX