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 m_f. So m_f(g) = f_g.

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) = ∂²_x(x³)₀ = (6x)₀ = 0.

(a) Prove that this form is symmetric and positive definite, and that the monomials xⁱy^j 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 m_f 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 a_i ≡ 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.