Minimal quadratic forms for the function field of a conic in characteristic 2

Abstract

In this note, we construct explicit examples of F Q F_Q -minimal quadratic forms of dimension 5 5 and 7 7 , where F Q F_Q is the function field of a conic over a field F F of characteristic 2 2 . The construction uses the fact that any set of n n cyclic p p algebras over a field of characteristic p p can be described using only n + 1 n+1 elements of the base field. It also uses a general result that provides an upper bound on the Witt index of an orthogonal sum of two regular anisotropic quadratic forms over a henselian valued field.