Honors Exam 2011 Algebra
1. Do either (i) or (ii).
(i) Let G denote the general linear group GL2(Fp) of invertible 2 × 2 matrices with entries in the field Fp of integers modulo a prime p. Find a Sylow p-subgroup of G, and determine the number of Sylow p-subgroups.
(ii) Prove that every group of order 48 contains a proper normal subgroup.
2. Let V be a finite-dimensional vector space with a skew-symmetric bilinear form ( , ), and let W be a two-dimensional subspace of V on which the restriction to W is nondegenerate. Explain why an orthogonal projection π: V → W exists, and find a formula for it, in terms of a suitable basis.
3. What facts about ideals in the integer polynomial ring Z[x] can one derive from the homomorphism Z[x]→ Z[i] that sends x to i?
4. Do either (i) or (ii).
(i) Decide whether or not the polynomial x4 +9x+9 generates a maximal ideal in the ring Q[x] of polynomials with rational coefficients.
(ii) Prove that every ideal I of the polynomial ring R = C[x1, ..., xn] that is not the whole ring is contained in a maximal ideal of R.
5. Do any two of the three parts.
(i) Let α1, α2, α3 denote the three complex roots of the polynomial f(x) = x3-x+1, listed in arbitrary order. Prove that αk1 + αk2 + αk3 is a rational number, for every positive integer k.
(ii) The notation is as in (i). To compute in the splitting field K = Q(α1, α2, α3) of f(x) using the symbols αi, one must be able to decide when two expressions in the roots are equal elements of K. Explain how this might be done.
(iii) Let K denote the field C(t) of rational functions in a variable t. This field has an automorphism σ that sends t to it-1. Determine the fixed field of the cyclic group < σ > generated by this automorphism.
6. Do either (i) or (ii).
(i) Gauss proved that a regular 17-gon can be constructed with ruler and compass. Explain what goes into this theorem, and prove as much as time permits.
(ii) Let V →f W be a linear transformation of real vector spaces, let k(f) denote the dimension of the kernel (the nullspace) of f, and let c(f) denote the dimension of the quotient space W/image(f). If k(f) and c(f) are finite, the index of f is defined to be the difference i(f) = k(f) - c(f). The index is not defined when k(f) or c(f) is infinite.
(a) Assuming that V and W are finite-dimensional, say dim V = m and dim W = n, what values are possible for the index i(f)?
(b) Let U →g V →f W be linear transformations. Prove that i(fg) = i(f)+i(g), provided that the terms are defined. If you can do so, prove this without assuming that the spaces U, V, W are finite-dimensional.