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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1999 • Question 11

Question

Math 6310 • Algebra Fall 1999 • Question 11

Let \(M\) be an \(n\) by \(n\) matrix with entries in \(\mathbb {C}\). Assume that there is a polynomial \(f(X)\) in \(\mathbb {Q}[X]\) which is irreducible over \(\mathbb {Q}\) such that \(f(M)=0\) (i.e. it is the zero matrix). Show that \(M\) is diagonalizable over \(\mathbb {C}\).

Raw LaTeX

Let \(M\) be an \(n\) by \(n\) matrix with entries in \(\mathbb{C}\). Assume that there is a polynomial
\(f(X)\) in \(\mathbb{Q}[X]\) which is irreducible over \(\mathbb{Q}\) such that \(f(M)=0\) (i.e. it is
the zero matrix). Show that \(M\) is diagonalizable over \(\mathbb{C}\).

Topic Tags

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