Minimal norm Jordan splittings of quadratic lattices over complete dyadic discrete valuation rings

Abstract

In [5], the so called minimal norm Jordan splitting over a ring of integers of dyadic local fields is introduced for determining the generators of integral orthogonal groups for the purpose of computing integral spinor norms. Such a normalization of Jordan splittings turns out to be useful in dyadic theory (see also [6] and [7]). In this note, we give a more conceptual proof and extend this result to a complete dyadic discrete valuation ring, where the residue field is not necessarily perfect. As an application, we discuss the Witt cancellation theorem and also give a proof of Theorem 10 in [4, Chapter 10], where the rigorous proof is not available. It should be pointed out that [3] gives some variation of the classification theorem over $\mathbb{Z}_2$ but not the detailed proof.