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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2019 • Question 10

Question

Math 6310 • Algebra Fall 2019 • Question 10

Suppose \(k\) is a field and suppose that \(f(x)\in k[x]\) has degree \(n\). Prove that \(f(x)\) has no repeated factors in \(k[x]\) if and only if all \(n\times n\) matrices over \(k\) with characteristic polynomial \(f(x)\) are similar.

Raw LaTeX

Suppose \(k\) is a field and suppose that \(f(x)\in k[x]\) has degree \(n\). Prove that \(f(x)\) has no
repeated factors in \(k[x]\) if and only if all \(n\times n\) matrices over \(k\) with characteristic
polynomial \(f(x)\) are similar.

Topic Tags

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