Algebra Qualifying Exam
March 19, 2011
Do all five problems.
1. Let Q denote the additive group of rational numbers.
(a) Prove that every finitely generated subgroup of Q is cyclic.
(b) Prove that Q is not finitely generated.
2. List, up to isomorphism, all abelian groups of order 3528 = 23 · 32 · 72 .
3. Let V be a vector space over a field F and let T : V → V be a linear
operator such that T 2 = T . Prove that V = ker(T ) ⊕ im(T ).
4. Let A, B be n × n matrices with entries in a field F and suppose that
AB = BA. Prove that if B has an eigenspace of dimension 1 then A
and B share a common eigenvector.
5. Let D be a principal ideal domain. Prove that every proper nontrivial
prime ideal is maximal.