Springer's theorem for tame quadratic forms over Henselian fields
A famous theorem by T.~A.~Springer states that the Witt group of quadratic forms over a complete discretely valued field of residue characteristic not \(2\) is isomorphic to a direct sum of two copies of the Witt group of the residue field. The case of residue characteristic \(2\) poses many subtle problems. The purpose of the present paper is to obtain a very general and characteristic free version of Springer's theorem for a certain subgroup of the Witt group which the authors call the Witt group of tame quadratic forms over a Henselian field \(F\) with valuation \(v\), denoted by \(I_{qt}(F)\). More precisely, let \(I_q(F)\) be the Witt group of even-dimensional nondegenerate quadratic forms over \(F\). Then \(I_{qt}(F)\) is the subgroup consisting of those classes of forms that become hyperbolic over a tame extension of \(F\), where a finite degree field extension \(E\) of the Henselian field \(F\) is called tame if it is defectless, with ramification degree not divisible by the characteristic of the residue field \(\overline{F}\) and its residue field \(\overline{E}\) is a separable extension of \(\overline{F}\). The paper's main theorem states that there is an isomorphism \(\partial: I_{qt}(F)\to \bigoplus_{\Gamma/2\Gamma}I_q(\overline{F})\) where \(\Gamma\) denotes the value group of \(v\), and where \(\partial\) is defined by residue forms for each coset of \(\Gamma\) modulo \(2\Gamma\) that depend on a choice of uniformizing parameter for the respective coset. This result in the case \(\text{char}(\overline{F})=2\) was essentially known and due to \textit{U.-P. Tietze} [Arch. Math. 25, 144--150 (1974; Zbl 0285.15018)], but his formulation and methods were quite different as he worked with the subgroup \(U(F)\) of \(I_q(F)\) generated by forms \(ax^2+bxy+cy^2\) with \(v(a)=v(b)=v(c)=1\) and used generators and relations for the Witt group. The authors show in the present paper that in fact \(U(F)=I_{qt}(F)\), thus recovering Tietze's result when \(\mathrm{char}(\overline{F})=2\). The authors also provide a further characterization of \(I_{qt}(F)\) in terms of quadratic forms that admit a certain type of norm called tame norm.
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