Ph.D. Preliminary Examination in Algebra

January 25, 1999

1. Let G be a finite group and Perm(G) the group of permutations of G viewed as a set.

(a) Show that the map

λ : G ⟶ Perm(G)

that is defined by λ(σ)(τ) = σ ∘ τ is a group homomorphism.

(b) Show that the map ρ_{1} defined by ρ_{1}(σ)(τ) = τ ∘ σ is a homomorphism if and only if G is an abelian group.

(c) Show that the map ρ defined by ρ(σ)(τ) = τ ∘ σ^{−1} is a homomorphism for every group G.

2. Let A be an n × n matrix in a field K, let c(t) be the characteristic polynomial of A, and let m(t) be the minimal polynomial of A. Show that m(t) divides c(t) in the polynomial ring K[t].

3. Show that the alternating group A_{4} has no subgroup of index 2.

4. Let f(x) = x^{5} − 2 in Q[x], and let K be the splitting field of f(x) over Q.

(a) Find generators for K as a Q-algebra.

(b) Find the Galois group G of K over Q.

(c) For each subgroup H of G describe the subfield of K which corresponds to H under the “funda-

mental correspondence of Galois theory”.

5. Show that if a finite ring R admits an injective (ring) homomorphism from a field, then the number of elements of R must be a power of a prime number.

6. Let R be a commutative ring, H a commutative R-algebra, and I an ideal in H. Show that

H/I ⊗_{R} H/I ≅ \frac{H ⊗_{R} H}{I ⊗_{R} H + H ⊗_{R} I} .

7. Let the field L be a (finite) Galois extension of the field K. Define tr : L → K by

tr(α) = ∑_{σ∈G} σ(α) .

Show that this trace map is surjective on K.

8. Let R be a ring and P a left R-module. Show that the following two statements are equivalent: (a) P is a direct summand of a finitely-generated free left R-module.

(b) There exist x_{1}, …, x_{n} ∈ P, and f_{1}, …, f_{n} ∈ Hom_{R}(P, R) such that the relation

x = ∑ f_{i}(x)x_{i}

holds for all x ∈ P.