Masters Comprehensive Exam - Modern Algebra - January 1995

1. List all Abelian groups of order 36 up to isomorphism (list each group exactly once).

2. Consider groups G of order 18.

(a) Give all primes p for which G has a p-Sylow subgroup.

(b) For each such p, give the order of the p-Sylow subgroups, and the possibilities for the number of p-Sylow subgroups.

(c) For each possible number of p-Sylow subgroups, give an example of a group G of order 18 with that number of p-Sylow subgroups, and list these subgroups in each example.

(d) Which of these subgroups are normal?

3. (a) Prove that an n × n real matrix A is diagonalizable if and only if R^{n} has a basis consisting of eigenvectors for A.

(b) Give an example of a 3 × 3 matrix that is not diagonalizable, and fully explain why not.

4. Let R be the ring consisting of all expressions a + bα + cβ + dγ, where a, b, c, d are complex numbers. Here addition is defined by

(a + bα + cβ + dγ) + (a′ + b′α + c′β + d′γ) =
(a + a′) + (b + b′)α + (c + c′)β + (d + d′)γ,

and multiplication is determined by the relations α^{2} = β^{2} = γ^{2} = −1, αβ = −βα = γ, βγ = −γβ = α, γα = −αγ = β, C-linearity and the distributive law. In other words, R is essentially the same as the quaternions, except that coefficients are complex rather than real, and α, β, γ have been used in place of i, j, k to avoid confusion with any i’s that occur in the complex coefficients a, b, c, d.

(a) Show that R is isomorphic as a ring to the ring of 2 × 2 complex matrices.

(b) Are the quaternions a division ring? Explain.

(c) Is R a division ring? Explain.

5. In this question, ω ∈ C denotes a primitive third root of unity. (a) Is Q( ^{3} \sqrt{2}) a Galois extension of Q? Explain.

(b) Find the splitting field K of the polynomial x^{3} − 2 over Q.

(c) Identify the Galois groups of K/Q and K/Q(ω) with more familiar groups.

(d) Find all intermediate fields between Q and K.