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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2014 • Question 4

Question

Math 6310 • Algebra Spring 2014 • Question 4

Let \(M\) be an \(n\times n\) matrix with entries from a field \(\mathbb {F}\) such that \(M^3=I\). Is \(M\) necessarily diagonalizable if (a) \(\mathbb {F}=\mathbb {Q}(\sqrt {3})\), (b) \(\mathbb {F}=\mathbb {Q}(i)\), (c) \(\mathbb {F}=\mathbb {F}_3\)?

Raw LaTeX

Let \(M\) be an \(n\times n\) matrix with entries from a field \(\mathbb{F}\) such that \(M^3=I\).
Is \(M\) necessarily diagonalizable if (a) \(\mathbb{F}=\mathbb{Q}(\sqrt{3})\),
(b) \(\mathbb{F}=\mathbb{Q}(i)\), (c) \(\mathbb{F}=\mathbb{F}_3\)?

Topic Tags

Field Extensions Finite Fields Ideals in Quadratic Integer Rings and Exterior Powers Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization