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 ρ1 defined by ρ1(σ)(τ) = τ ◦ σ is a homomorphism if and only if G is an
(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].
Show that the alternating group A4 has no subgroup of index 2.
4. Let f(x) = x5 − 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”.
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 ∼=
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
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 x1, . . . , xn ∈ P , and f1, . . . , fn ∈ HomR(P,R) such that the relation
holds for all x ∈ P .