SPRINGER'S ODD DEGREE EXTENSION THEOREM FOR QUADRATIC FORMS OVER SEMILOCAL RINGS

Philippe Gille (1) , Erhard Neher (2)

1 ICJ - Institut Camille Jordan
2 Department of Mathematics and Statistics [Ottawa]

A fundamental result of Springer says that a quadratic form over a field of characteristic not 2 is isotropic if it is so after an odd degree extension. In this paper we generalize Springer's theorem as follows. Let R be a an arbitrary semilocal ring, let S be a finite R-algebra of odd degree, which is étale or generated by one element, and let q be a nonsingular R-quadratic form whose base ring extension q S is isotropic. We show that then q is already isotropic.

Mots clés: Quadratic forms; semilocal rings; Springer Odd Degree Extension Theorem; Springer Odd Degree Extension Theorem MSC 2000: 11E08; 11E81

Domaines: Anneaux et algèbres [math.RA]

Philippe Gille, Erhard Neher. SPRINGER'S ODD DEGREE EXTENSION THEOREM FOR QUADRATIC FORMS OVER SEMILOCAL RINGS. 2021. ⟨hal-03124810v1⟩