Pythagorean *-fields

Abstract

We generalize the usual notion of the pythagorean field (every sum of squares is a square) to the setting of division rings D with involution. We choose a definition that works particularly well with Baer orderings and Witt groups of hermitian forms.

Valuations and Hermitian forms on skew fields

“…The main efforts to carry over the reduced theory of quadratic forms to * -fields, viewing forms as functions on a topological space of orderings, have been by Craven and Smith [11,15,16], though an involution was introduced long ago into the commutative theory by Knebusch, Rosenberg and Ware [26]. A hermitian form over (D, * ) can be diagonalized to have the form φ = a 1 , a 2 , . .…”