Algebra Prelim, August 2012

Do seven problems

1. Let G be a simple group of order 4312 = 2³·7²·11.

   (a)

   Prove that G has a subgroup of order 77 (consider the number of Sylow 11-subgroups).

   (b)

   Prove that G has a subgroup of order 7 whose normalizer contains a group of order 49 and a group of order 77.

   (c)

   Prove that no such G exists.

2. Let R be a UFD and let I₁⊆I₂⊆...⊆R be an ascending chain of principal ideals of R. Prove there exists a positive integer N such that I_n = I_N for all n≥N.

3. Let q∈ℤ be a prime and let P be a nonzero projective ℤ-module (not necessarily finitely generated).

   (a)

   Prove that P≠Pq.

   (b)

   Prove there exists a ℤ-module epimorphism P↠ℤ/qℤ (you may assume that every subspace of a vector space has a direct complement).

   (c)

   Prove that P⊗_ℤP≠ 0.

4. Let M and N be finitely generated ℤ-modules. Suppose M is isomorphic to a submodule of N and N is isomorphic to a submodule of M. Prove that M≌N.

5. Let d, n be a positive integers and let A∈M_d(ℂ). Suppose Aⁿ = 0. Determine the characteristic polynomial of A and prove that A^d = 0.

6. Let ζ = e^2πi/13, a primitive 13 th root of unity. Prove that ℚ(ζ) contains exactly one subfield K such that [K : ℚ] = 6. Prove further that K is a Galois extension of ℚ and that K⊂ℝ.

7. Let k be a field, let d be a positive integer, and let S⊆k^d. Let f₁, f₂,... be a sequence of polynomials in k[x₁,..., x_d] with the property that for every s∈S, there exists a positive integer n such that f_n(s)≠ 0. Prove that there exists a positive integer N such that for every s∈S, there exists n≤N such that f_n(s)≠ 0.

8. Compute the character table of S₃×ℤ/2ℤ.