Algebra Prelim

June 1997

Provide proofs for all statements.

If necessary, you may use the results from other parts of this test, even though you may not have successfully proved them.

1. Prove that ZZ/(4) ⊕ ZZ/(4) is not isomorphic to any subgroup of ZZ/(16) ⊕ ZZ/(2).

2. Let V be a finite dimensional vector space over a field k. Let f : V ⟶ V be a linear transformation such that f ∘ f = f. Prove that V has a basis v_{1}, v_{2}, ⋯, v_{r}, f(v_{r+1}), f(v_{r+2}), ⋯, f(v_{n}) where f(v_{1}) = f(v_{2}) = ⋯ = f(v_{r}) = 0.

3. Let G be a group and K < H < G be subgroups such that K is a direct factor of H and H is a direct factor of G. Prove that K is a normal subgroup of G and that

G/K ≅ H/K × G/H

4. Let K ⊂ L be fields of characteristic 0 such that [L : K] = n < ∞. Prove that there are at most n homomorphisms σ : L ⟶ L such that σ(a) = a for all a ∈ K.

5. If S ⊂ R is a multiplicative set of a commutative ring R, argue that for any a ∈ R, the element a/1 is a unit of S^{−1}R if and only if ac ∈ S for some c ∈ R.

6. Let k be a field and let f_{1}(X), f_{2}(X) be separable polynomials over k with splitting fields E_{1}, E_{2} respectively. We assume that all these fields are inside some fixed algebraic closure of k. Let F denote the compositum of E_{1}, E_{2} (i.e. F is the smallest field containing both E_{1}, E_{2}).

• Show that there are natural homomorphisms

θ_{i} : Gal(F, k) ⟶ Gal(E_{i}, k)

• The induced map

θ : Gal(F, k) ⟶ Gal(E_{1}, k) × Gal(E_{2}, k)

defined by θ(σ) = (θ_{1}(σ), θ_{2}(σ)) is injective.

• If the field degrees [E_{1} : k] and [E_{2} : k] are coprime, then θ is an isomorphism.

7. Let R = Q [X]/(X^{2}+1). Is R an integral domain? Is it a field? Let S = Q [X, Y]/(X^{2}+1, Y^{2}+1). Is S an integral domain? Is it a field?

8. What is the degree of the splitting field of X^{6} − 3X^{3} − 4 over Q ? What is the degree of the splitting field of the same polynomial over ZZ_{3} = ZZ/(3)?

9. Prove that an irreducible element in a UFD generates a prime ideal. Let v, w be two irreducible elements in a UFD, such that the ideal I = (v, w) is not the unit ideal. Give an example to show that I need not be a prime ideal.

10. Let G be a finite group and let a ∈ G. Let S be the conjugacy class of a. Prove that the number of elements in S divides the order of the group G.