Honors Exam 2010 Algebra
1. Let G be a nonabelian group of order 28 whose Sylow 2 subgroups are cyclic.
(a) Determine the numbers of Sylow 2-subgroups and of Sylow 7-subgroups.
(b) Determine the numbers of elements of G of each order.
(c) Prove that there is at most one isomorphism class of such groups.
2. This problem is about the space V of real polynomials in two variables x, y. The fact that it is an infinite-dimensional space plays a role only in part (c).
If f is a polynomial, ∂f will denote the operator f(∂/∂x, ∂/∂y), and ∂f(g) will denote the result of applying this operator to a polynomial g. We also have the operator of multiplication by f, which we write as mf. So mf(g) = fg.
The rule (f, g) = ∂f(g)0 defines a bilinear form on V , the subscript 0 denoting evaluation of a polynomial at the origin. For example, (x2, x3) = ∂2x(x3)0 = (6x)0 = 0.
(a) Prove that this form is symmetric and positive definite, and that the monomials xiyj form an orthogonal basis of V (not an orthonormal basis).
(b) Linear operators A and B on V are adjoint if (Ap, q) = (p, Bq) for all polynomials p and q. Prove that ∂f and mf are adjoint operators.
(c) When f = x2 + y2, the operator ∂f is the Laplacian, which is often written as ?. A polynomial h is harmonic if ?h = 0. Let be the space H of harmonic polynomials. Identify the space H⊥ orthogonal to H with respect to the given form.
3. Do either (a) or (b).
(a) Describe the maximal ideals in Z[x].
(b) Determine the number of irreducible polynomials of degree 4 in F3[x].
4. Do either (a) or (b).
(a) Let ?: Z[x] → C be the homomorphism that sends x to a complex number γ. Prove that the kernel of ? is a principal ideal.
(b) Let f(x) = x5 + cx4 + a3x3 + a2x2 + a1x + a0 be an integer polynomial such that ai ≡ 0 modulo 3 for all i and that a0 
0 modulo 9. There is no condition on the coefficient c of x4 other than that it is an integer. Prove that f is irreducible in Z[x] unless it has an integer root.
5. Do any two of the three parts.
Let R denote the ring of Gauss integers: R = {a + bi | a, b ∈ Z}.
(a) A Gauss prime is a Gauss integer that has no proper Gauss integer factor and is not a unit in R. Factor 3 + 9i into Gauss primes.
(b) Let M denote the additive group (Z/5Z)+. In how many ways can M be given the structure of an R-module?
(c) Solve AX = B for X in R2, when
6. Let α = 3√2, β = √3, and γ = α + β. Let L be the field Q(α, β), and let K be the splitting field of the polynomial (x3 - 2)(x2 - 3) over Q.
(a) Determine the degrees [L: Q] and [K: Q].
(b) Determine all automorphisms of the field L.
(c) Determine the degree of γ over Q.
(d) Let f be the irreducible polynomial for γ over Q. What are the complex roots of f?
(e) Determine the Galois group of K/Q.