Ph.D. Preliminary Examination
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
Fp 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
6= 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
σ(α). Let Tα : L→ L be the K-linear transformation defined by Tα(β) =
αβ. Show that tr(α) is the trace of the linear transformation Tα.
6. Prove that for any prime p, there are at least four isomorphism classes of groups of
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
(b) Give an example of a non-flat Z-module.
8. Let K be a field, M a K-vector space. Let M∗ = HomR(M,K). Show that the
canonical map M →M∗∗ is surjective if and only if M is finite dimensional.