Submission: 1998, Aug. 22
If $F$ is a field with a (Krull) valuation, then the filtration of $F$ induced by the valuation yields an associated graded ring, which is a graded field. Conversely, if $R$ is a graded field with totally ordered grade group, then $R$ is an integral domain and there is a canonically associated valuation on the quotient field of $R$. The processes of passing from valued field to graded field and vice versa are not quite inverses of each other, but many properties in one setting are well-reflected in the other.
The goal of this paper is to describe an algebraic extension theory for graded fields analogous to what is known for valued fields, and then to spell out the correspondence between tame extensions of graded fields and Henselian valued fields. This has the benefit that graded fields are easier to work with for many purposes than valued fields. But beyond this, there is a similar correspondence between graded division rings and valued division rings, where the graded objects seem to be significantly easier to work with than the valued objects. We first learned of this correspondence from a paper by M\. Boulagouaz [B$_2$]. The correspondence for division rings is actually far more extensive than what was described by Boulagouaz, and we pursue that subject in a sequel to this paper [HW]. The choice of topics to treat here was influenced by the needs of the study of division rings. But, we feel that the commutative theory presented here is of interest in its own right.
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