On the dimensions of quadratic forms.
According to Arason-Pfister's Hauptsatz the dimension of an anisotropic quadratic form in the \(n\)-th power \(I^n(k)\) of the fundamental ideal \(I(k)\) of the Witt ring \(W(k)\) is at least \(2^n\). The main result of the present paper is that, for any field \(k\) of characteristic zero, if the dimension of an anisotropic form in \(I^n(k)\) is greater than \(2^n\), then it is at least \(2^n + 2^{n-1}\). The proof relies on a Main Lemma whose proof, only roughly sketched, depends on techniques developed by the author in an earlier preprint, and on Voevodsky's work leading to the proof of Milnor's Conjecture. The Main Lemma is the following statement on forms in the powers \(J^n(k)\) of the Knebusch ideal \(J(k)\) over a field \(k\) of characteristic zero. Let \(p\) and \(q\) be anisotropic forms in \(J^n(k)\) of height \(2\) and dimension less that \(2^{n+1}\). Suppose \(p\) is isotropic over the function field \(k(q)\) and \(q\) is isotropic over the function field \(k(p)\). Then for any field extension \(K \supset k\), the form \(p\) is hyperbolic over \(K\) if and only if \(q\) is hyperbolic over \(K\). An application of the techniques used to prove the main result gives also the complete list of dimensions and splitting patterns for all forms of height \(2\) over a field of characteristic zero. The restriction to fields of characteristic zero comes from its presence in Voevodsky's constructions. The main result of this paper has been generalized to fields of any characteristic different from two by \textit{N. Karpenko} and \textit{A. Merkurjev} [Invent. Math. 153, No. 2, 361--372 (2003; Zbl 1032.11015)].
scientific article; zbMATH DE number 2074632