A galois approach to mth roots of matrices with rational entries Original Research Article

Let A be a given n × n matrix with rational entries and irreducible characteristic polynomial f(x). We investigate the Galois groups of f(x) and f(xm), to find necessary and sufficient conditions for the existence of a solution B to the matrix equation A = Bm, where B is also a matrix with rational entries. We do this by finding necessary and sufficient conditions for f(xm) to have a factor of degree n (with rational coefficients).