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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.