On the 3-Pfister Number of Quadratic Forms

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F^×2-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse–Witt invariant can be written in the Witt ring as the sum of at most (d²)/8 3-fold Pfister forms.