On 14-dimensional quadratic forms in \(I^3\), 8-dimensional forms in \(I^2\), and the common value property
The authors study several properties of fields expressed in terms of the behavior of higher-dimensional quadratic forms. The difference of two 3-fold Pfister forms over a field \(F\) belongs to \(I^3(F)\), the third power of the fundamental ideal of the Witt ring \(W(F)\), and the authors say a field \(F\) satisfies \(D(14)\) if each 14-dimensional quadratic form in \(I^3(F)\) is similar to the difference of two 3-fold Pfister forms. The property \(D(8)\) requires that each form in \(I^2(F)\) of dimension 8 with Clifford invariant represented by a biquaternion algebra is the sum of two forms similar to 2-fold Pfister forms, which reverses another generally valid implication. \(D(4)\) holds for \(F\) when, for every pair of 4-dimensional quadratic forms with the same signed determinant, if the forms become equivalent over a quadratic extension field of \(F\), then they contain subforms similar over \(F\). Finally, \(D(2)\) is a common value property studied earlier in the literature: if two binary forms over \(F\) represent a common value over a quadratic extension \(E\) of \(F\), then they represent over \(E\) a common value in \(F\). The authors establish that \(D(2)\) implies \(D(4)\), and also \(D(8)\) implies \(D(14)\). They show that the implications cannot be reversed in general. It is not known whether \(D(4)\) implies \(D(8)\). The authors study in detail the behavior of the properties \(D(n)\) when passing from a field \(K\) with a 2-henselian discrete valuation to the residue class field \(k\), or from \(k\) to \(K\). For example, they establish that if \(D(2n)\) for \(n=2,4\) (or \(D(14)\)) holds for \(K\), then \(D(n)\) (or \(D(8)\), resp.) holds for \(k\). Also \(D(2)\) lifts from \(k\) to \(K\), \(D(4)\) and \(D(8)\) for \(k\) imply \(D(8)\) for \(K\), and \(D(8)\) for \(k\) implies \(D(14)\) for \(K\). These results are then used to give a series of new examples of fields having or failing to have the properties \(D(n)\). The fields of (iterated) formal power series appear quite often. Thus, for instance, the iterated formal power series fields over global or local fields all have the properties \(D(n)\). More generally, power series fields with coefficient field \(F\) having the Hasse number \(< 14\) are considered. The final section of the paper discusses the relations between \(D(n)\) and the excellence property of an extension field, and gives an explicit and systematic list of counterexamples to each \(D(n)\) for the rational function fields in \(\leq 4\) variables over the rational numbers.
excellence property of an extension field
scientific article; zbMATH DE number 1213152