Algebra Prelim, August 2018

Do all problems

  1. Let G be a group order 495 = 9·5·11. Prove that G has a nontrivial normal Sylow subgroup. Deduce that G has a normal subgroup of order 9.

  2. Let 0≠I⊲ℤ[x] and let n be the lowest degree of a nonzero polynomial in I. Suppose I contains a monic polynomial of degree n. Prove that I is a principal ideal.

  3. Let I be an injective ℤ-module. Prove that ℚ/ℤ⊗ℤI = 0.

  4. Let R be a PID, let M be a finitely generated R-module, and let C be a cyclic submodule of M. Prove that there exists an R-epimorphism M↠C (first consider the case when M is not a torsion module).

  5. Let K/ℚ (where K⊂ℂ) be a finite Galois extension with Galois group A5 and let R = K∩ℝ. Suppose R≠K.
    1. Prove that K is the splitting field of an irreducible polynomial f∈ℚ[x] of degree 6 (you may assume that A5 has a subgroup of order 10).

    2. Prove that [R : ℚ] = 30.

    3. How many real roots does f have?

    4. Prove that f has roots a, b such that ℚ(a, b) = R.

  6. Determine the possible rational canonical forms for a 3×3 matrix over 𝔽2 (the field with two elements) which has trace and determinant 1.

  7. Compute the character table of A4. (You may assume that A4 has 4 conjugacy classes with representatives (1), (1 2 3), (1 3 2), (1 2)(3 4), and that A4 has a normal subgroup of order 4.)