8 - Equations and singularities

Summary

The theory of equations over skew fields is in many ways the least developed part of the subject. An element of an extension field whose powers are (right) linearly dependent over a subfield satisfies a rather special kind of equation (discussed in 3.4). More general equations are much less tractable, and in some ways it is more appropriate to consider singularities of matrices, a wider problem. Here we can limit ourselves to linear matrices, but even in this case there is as yet no comprehensive theory.

We begin by discussing the different possible notions of an algebraically closed skew field; although their existence in all cases has not yet been established, the relations between them are described in 8.1. We find that to solve equations we need to find singular eigenvalues of a matrix and this has so far been done only in special cases (discussed in 8.5). By contrast the similarity reduction to diagonal form requires the notion of left and right eigenvalues; it is shown in 8.2 that they always exist (in a suitable extension field) and 8.3 describes the reduction to normal form based on these eigenvalues. While this normal form (over an EC-field) is quite similar to the commutative case, there is no full analogue of the Cayley–Hamilton theorem (owing to the lack of a determinant function), although such a result exists for ‘skew cyclic’ matrices (i.e. matrices A such that xI - A is stably associated to a 1 × 1 matrix) and is presented in 8.3.