A Galois approach to \(m\)th roots of matrices with rational entries
Let \(A\in M_n(\mathbf Q)\) be a matrix with irreducible characteristic polynomial \(f(x)\). The author investigates necessary and sufficient conditions for the existence of a solution \(B\in M_n(\mathbf Q)\) such that \(A=B^m\). By a previously unpublished proposition, this matrix equation has solutions in \(M_n(\mathbf Q)\) if and only if \(f(x^m)\) has a factor of degree \(n\) in \(\mathbf Q[x]\). Then the author investigates the Galois groups of \(f(x)\) and \(f(x^m)\) to find the necessary and sufficient conditions for \(f(x^m)\) to have a factor of degree \(n\) in \(\mathbf Q[x]\). The results are as follows: Under the additional assumptions that \(m\) is odd and that \(\mathbf Q(\lambda_1,\dots,\lambda_n)\cap\mathbf Q(e^{2\pi i/m})=\mathbf Q\) where \(\lambda_i\) are \(n\) roots of \(f(x^m)\) such that \(\lambda_i^m\) are all the roots of \(f(x)\), a necessary and sufficient condition for \(A=B^m\) has a solution in \(M_n(\mathbf Q)\) is \(|G(K/\mathbf Q)|= \phi(m)|G(L/\mathbf Q)|\) where \(\phi\) is Euler's \(\phi\)-function, \(K\) or \(L\) is the splitting field of \(f(x^m)\) or \(f(x)\) over \(\mathbf Q\) respectively. If \(m\) is an odd prime, then \(f(x^m)\) has a factor of degree \(n\) in \(\mathbf Q[x]\) if and only if the Galois group \(G(K/L)\) is abelian.
scientific article; zbMATH DE number 1026438