Linkage of Pfister forms over semi-global fields
Let \(K\) be a field and \(d\) be a natural number. The \(d\)-fold Pfister forms over \(K\) form a natural class of quadratic forms in \(2^d\) variables over \(K\). We say that \(d\)-fold Pfister forms are linked, if any pair of \(d\)-fold Pfister forms over \(K\) contains a common \((d-1)\)-fold Pfister form.\N\NThe paper under review contains a new proof of the fact, that three-fold Pfister forms over a function field in one variable over a \(p\)-adic field are linked. This approach relies on an abstract study of how linkage properties for quadratic forms over function fields can be lifted from residue fields of henselian valued fields to the valued fields themselves. Iterating this process we get the following theorem: \N\NLet \(d\) be a non-zero natural number and suppose there is a sequence \(K_1,K_2,\ldots,K_d\) of fields, where \(K_1\) is finite, and each \(K_{i+1}\) is complete with respect to a discrete valuation with residue field \(K_i\). For any function field in one variable \(F/K_d\), \((d+1)\)-fold Pfister forms are linked. As an application we obtain that \((d+2)\)-fold Pfister forms over function fields in one variable over a \(d\)-dimensional higher local field are linked.
scientific article; zbMATH DE number 7930936
scientific article