Maybe I once learned something about this, but if so, I can't recall now: Consider a collection M of matrices over a ring, like M = M(n, Z) or M = M(n, Q) or M = M(n, R) or M = M(n, C), meaning the ring of n x n square matrices over the integers, the rationals, the reals, or the complexes. For any one of these, call it M and consider polynomials of form P(X) = X^k + A_(k-1) X^(k-1) + ... + A_1 X + A_0 where the A_j belong to the matrix ring M. Then what is known a) about the existence of solutions X in M to the equation P(X) = 0 (where 0 denotes the 0 matrix in M) ??? b) about closed formulas for the roots of P(X) in M ??? (like the quadratic, cubic, and quartic formulae). Of course, there can't be any general quintic formula* for matrices, since that would contradict Abel's theorem on its unsolvability over C. —Dan ————————————————————————————————————————————————————————————————————— * Allowing the four arithmetic operations as well as rational powers.