Matroidal root structure of skew polynomials over finite fields

A skew polynomial ring R = K [x; sigma, delta] is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, evaluations, and roots. A polynomial in such a ring may have more roots than its degree, which leads to the concepts of closures and independent sets of roots. In R = F-qm[x, sigma], this leads to the matroids m and m of right independent and left independent sets, which are isomorphic via the extension of the map phi : [1] -> [1] defined by phi (a) = a(m), where i = q(i-1) -1/q-1. Extending the field of coefficients of R results in a new ring S of which R is a subring, and if the extension is taken to include roots of an evaluation polynomial of f (x), then all roots of f (x) in S are in the same conjugacy class.