Springer’s Theorem for Tame Quadratic Forms over Henselian Fields

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tamely ramified extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2. A celebrated theorem of Springer [8] establishes an isomorphism between the Witt group of a complete discretely valued field and the direct sum of two copies of the Witt group of the residue field, provided the residue characteristic is different from 2. Springer also considered the case where the residue characteristic is 2, and he pointed out the extra complications that arise from the fact that the residue forms are not necessarily nonsingular, even when the residue field is perfect. The exhaustive analysis by Aravire–Jacob [1] shows that the description of the Witt group of a dyadic Henselian field is extremely delicate, even when the field is maximally complete of characteristic 2. Our purpose is to show that, by contrast, Springer’s original techniques—as revisited in [7]—yield a very general characteristic-free version of Springer’s theorem for the tame part of the Witt group. Our main result is the following: let F be a field with a Henselian valuation v, with value group ΓF and residue field F of arbitrary characteristic; there is a group isomorphism