PhD COMPREHENSIVE EXAM – ALGEBRA – JUNE 1993

1. Determine all groups of order 117 up to isomorphism.

2. Let R be a commutative ring with identity. For any ideal 𝔞 of R, let

V (𝔞) = {𝔭 | 𝔭 is a prime ideal of R with 𝔞 ⊂ 𝔭}
rad(𝔞) = {x ∈ R | x^{n} ∈ 𝔞 for some integer n ≥ 1}

Show that for any two ideals 𝔞 and 𝔟 of R, V (𝔞) ⊂ V (𝔟) iff rad(𝔞) ⊃ rad(𝔟). (Hint: There is an equivalent description of rad(𝔞). Prove it and use it.)

3. Prove that an abelian group T is divisible iff it is injective as a ℤ-module.

4. Prove or disprove each of the following statements. Proofs should be from the definitions.

(a) The polynomial ring ℤ[X_{1}, …, X_{n}] in a finite number of variables is a Noetherian ring.

(b) The polynomial ring ℤ[X_{1}, X_{2}, …] in an infinite number of variables is a Noetherian ring.

5. Let F be a finite field.

(a) Prove that its multiplicative group F^{×} is cyclic.

(b) Prove that F is perfect.

6. Compute the Galois group of the polynomial f(x) = x^{3} − 2x + 2 over each of the following fields: (a) F = ℚ, (b) F = ℝ, (c) F = 𝔽_{5}. In each case describe the lattice of subfields between the ground field F and the splitting field E of f(x).

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