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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2018 • Question 7

Question

Math 6310 • Algebra Spring 2018 • Question 7

Prove that if a \(3\times 3\) matrix \(A\) over \(\mathbb {Q}\) satisfies \(A^8=I\), then \(A^4=I\). Justify claims of irreducibility of polynomials.

Raw LaTeX

Prove that if a \(3\times 3\) matrix \(A\) over \(\mathbb{Q}\) satisfies \(A^8=I\), then \(A^4=I\).
Justify claims of irreducibility of polynomials.

Topic Tags

Galois Groups of Polynomials Irreducible Polynomials Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization Polynomial Rings and Ring Homomorphism Kernels