Algebra Prelim, August 2012
Do seven problems
- Let G be a simple group of order
4312 = 23·72·11.
- Prove that G has a subgroup of order 77 (consider the number of
- Prove that G has a subgroup of order 7 whose normalizer contains a
group of order 49 and a group of order 77.
- Prove that no such G exists.
- 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 In = IN for all
q∈ℤ be a prime and let P be a
ℤ-module (not necessarily finitely generated).
- Prove that
- Prove there exists a
P↠ℤ/qℤ (you may assume that every
subspace of a vector space has a direct complement).
- Prove that
- Let M and N be finitely generated
Suppose M is isomorphic to a submodule of N and N is isomorphic
to a submodule of M. Prove that
- Let d, n be a positive integers and let
Suppose An = 0. Determine the characteristic polynomial of A
and prove that Ad = 0.
ζ = e2πi/13, a primitive 13 th root of unity. Prove
ℚ(ζ) contains exactly one subfield K such that
[K : ℚ] = 6. Prove further that K is a Galois extension
ℚ and that
- Let k be a field, let d be a positive integer, and let
f1, f2,... be a sequence of polynomials in
k[x1,..., xd] with the property that for every
exists a positive integer n such that
fn(s)≠ 0. Prove that
there exists a positive integer N such that for every
n≤N such that
- Compute the character table of