Ph.D. Preliminary Examination ALGEBRA Fall 1995

1. Let K be a field, L a field extension of K. An element α in L is algebraic over K if α is the root of some monic polynomial with coefficients in K. Show that if α and β in L are algebraic over K, then αβ is algebraic over K.

2. Let w be a primitive cube root of unity. Let R = Z[w]. Let λ = 1 − w. Show that R/λR ≅ Z/3Z.

3. Let G be the group of 2 × 2 invertible matrices of determinant 1 with coefficients in the field of 3 elements.

(a) Show that G has order 24.

(b) Find the number of 3-Sylow subgroups of G.

4. Let G be a finite p-group, p prime, V a finite dimensional vector space over the field F_{p} of p elements. Suppose G acts linearly on V (i.e. there is a homomorphism from G into the group GL(V) of invertible linear transformations from V to V). Prove that G has a non-zero fixed point: that is, there is some α ≠ 0 in V so that σ(α) = α for all σ in G.

5. Let L/K be a Galois extension of fields with Galois group G. Let L = K[α]. Define tr(α) = ∑_{r∈G} σ(α). Let T_{α} : L → L be the K-linear transformation defined by T_{α}(β) = αβ. Show that tr(α) is the trace of the linear transformation T_{α}.

6. Prove that for any prime p, there are at least four isomorphism classes of groups of order p^{3}.

7. A Z-module M is flat if for any short exact sequence 0 → A → B → C → 0 of Z-modules, the sequence 0 → M ⊗ A → M ⊗ B → M ⊗ C → 0 is exact.

(a) State and prove a criterion for flatness as follows: M is flat if and only if for any

homomorphism f : E → F of Z-modules, if t is jective, then M ⊗ f is

jective.

(b) Give an example of a non-flat Z-module.

8. Let K be a field, M a K-vector space. Let M^{∗} = Hom_{R}(M, K). Show that the canonical map M → M^{∗∗} is surjective if and only if M is finite dimensional.