The minimal height of quadratic forms of given dimension
The height \(h(\phi)\) of an anisotropic quadratic form over a field is the length of the splitting tower of \(\phi\). \textit{J. Hurrelbrink} and \textit{U. Rehmann} [J. Reine Angew. Math. 444, 183--192 (1993; Zbl 0791.11016)] determined the height of an excellent form of dimension \(n\) from the representation of \(n\) in the form of alternating sum of distinct \(2-\)powers. They conjectured that the height of anisotropic excellent form of dimension \(n\) is minimal among the heights of all anisotropic forms of dimension \(n\). In the present paper this conjecture is proved by using an improved version of the results of \textit{N. A. Karpenko} [Invent. Math. 153, No.~2, 455--462 (2003; Zbl 1032.11016)].
splitting pattern of anisotropic quadratic forms
scientific article; zbMATH DE number 5126583