Normal subgroups can be used to decompose groups, and this is an important tool in the analysis of groups. Ideals play a similar role in ring theory, but there is no direct analogue in fields. The nearest equivalent is a general valuation, which allows a field to be analysed into a group, the ‘value group’, and a residue-class field. Thus valuations form a useful tool in commutative field theory, but there is no method of construction in general use, mainly because in most cases all the valuations are explicitly known, e.g. for algebraic number fields, function fields of one variable or even two variables (see e.g. Cohn [91], Ch. 5). Our aim in this chapter is to describe a general method of construction, using subvaluations, which can be used even in the non-commutative case.

We begin by recalling the basic notions in 9.1, which still apply to skew fields, and then in 9.2 explain the special case of an abelian value group, which presents a close analogy to the commutative case while being sufficiently general to include some interesting applications. In the commutative case a ring R with a valuationv is an integral domain and v extends in a unique way to the field of fractions of R. In the general case neither existence nor uniqueness is ensured; what is needed here is a valuation on all the square matrices over R and 9.3 introduces the study of such matrix valuations and explains the way they determine valuations on epic R-fields.