PH. D. COMPREHENSIVE EXAM ALGEBRA JANUARY, 1995

Notation: Z_{n} denotes the integers modulo n.

1. (It is permissible to use the Sylow theorems.)

(a) Show that every group of order p^{n}, where p is a prime number, is solvable.

(b) Show that every group of order p^{2}q^{2}, where p and q are prime numbers with p > q ≥ 3, is solvable.

2. Explicitly construct a subgroup H of G = Z_{27}⊕Z_{3} such that G/H ≃ Z_{9} or show that no such subgroup exists.

3. Let M be a Noetherian module over the ring R and let ϕ : M → M be a surjective R-module homomorphism. Show that ϕ is an isomorphism.

4. Let R be a ring with identity 1 ≠ 0. Let J be the intersection of all the maximal left ideals of R. Show that x ∈ J if and only if 1 − rx is a unit of R for all r ∈ R.

5. Factor the polynomial x^{4} − 2x^{2} − 15 into irreducible factors over the fields Q, R, C, Z_{7}, Z_{19} and compute its Galois group over each of these fields.

6. Compute the Galois group of x^{4} + 2 over Q.