Thanks for the paper suggestions, Victor! It occurs to me that as long as the matrices are over a ring M (assumed to have a multiplicative identity) in which the following hold: ----- a) 2 has a multiplicative inverse ...* —and— b) every element of Ring has a square root in Ring ... ----- ... then a quadratic equation of form X^2 + b X + c for b, c in Ring and the unknown X to be found within Ring as well ... ... has a solution (OK, two) given by the usual formula: ----- X = -b/2 ± √((b/2)^2 - c). ----- I'm guessing the cubic and quartic formulae for C work, too, for rings, that have some additional restrictions regarding characteristic, at least. (Cubics should definitely *not* be characteristic = 3 or we may have trouble dividing by 27.) But commutativity may be a problem for degrees 3 and 4. —Dan ————— * Here 2 denotes 1 + 1, in case you were wondering. I wrote: ----- ... 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. -----