Springer's theorem for tame quadratic forms over Henselian fields

TL;DR

This paper proves a theorem relating tame quadratic forms over Henselian fields to graded quadratic forms, establishing an isomorphism between their Witt groups and providing a structural understanding of these forms.

Contribution

It introduces a canonical isomorphism between the Witt group of tame quadratic forms over Henselian fields and the Witt group of graded quadratic forms, extending classical results.

Findings

Witt group of tame quadratic forms is isomorphic to that of graded quadratic forms.

The Witt group decomposes into a direct sum indexed by the value group modulo 2.

The theorem applies to fields with arbitrary residue characteristic.

Abstract

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.