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Unformatted text preview: Algebra Qualifying Examination January, 2003 Directions: 1. Answer all questions. (Total possible is 100 points.) 2. Start each question on a new sheet of paper. 3. Write only on one side of each sheet of paper. Policy on Misprints: The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem in such a way that it becomes trivial. Notes: 1. All rings are unitary. All modules are unitary. 2. is the rationals, the reals, the complexes, and the integers. Problems 1. (10 points) Let T : 3 → 3 be a linear transformation. Prove that T has a one dimensional invariant subspace and a two dimensional invariant subspace. 2. (10 points) Let G = / . Prove that, for all t ∈ + , G has a unique cyclic subgroup of order t ....
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- Spring '08
- Prime number, Integral domain, dimensional invariant subspace