Oh, well. I knew there was some reason I didn't become an algebraist. (I wanted to look at one-sided polynomials because I thought they might be easier to deal with than general polynomials. The same problem — non-commutativity — seems to be what prevents there being a good theory of analytic functions with quaternion or octonion coefficients.) —Dan Andy Latto wrote: ----- I think this doesn't work because of non-commutativity. You can't factor X^2 + 2DX + D^2 as (X + D)^2 ( a necessary step in deriving the quadratic formula), because the latter expands to X^2 + DX + XD + D^2, which is different. Which raised the question of why the polynomials with the coefficients before the variables are the interesting ones. Shouldn't you be looking at the more general quadratic X^2 + AX + XB + C = 0? I wrote:
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).