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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Summer 2020 • Question 3

Question

Math 6310 • Algebra Summer 2020 • Question 3

Suppose \(M\) is a \(3\times 3\) matrix, with entries from the field of real numbers, such that \[ M^8=I\qquad \text {and}\qquad M^4\ne I. \] Determine the possibilities for the minimal polynomial of \(M\).

Raw LaTeX

Suppose \(M\) is a \(3\times 3\) matrix, with entries from the field of real numbers, such that
\[
M^8=I\qquad\text{and}\qquad M^4\ne I.
\]
Determine the possibilities for the minimal polynomial of \(M\).

Topic Tags

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