The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

TL;DR

This paper extends the Artin-Springer theorem to semi-local rings with finite residue fields, showing anisotropic quadratic forms remain anisotropic over odd-degree finite tale extensions, broadening applicability in algebraic geometry.

Contribution

It generalizes the classical Artin-Springer theorem from fields to semi-local rings with finite residue fields, enabling new isotropy criteria for quadratic spaces.

Findings

Anisotropic quadratic spaces over such rings stay anisotropic after odd-degree finite tale extensions.

The result applies to rings with at least one finite residue field, expanding previous infinite residue field cases.

Extension of isotropy criteria to broader classes of semi-local domains.

Abstract

Let $R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $R$ remain anisotropic after base change to any odd-degree finite \'{e}tale extension of $R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $R$ is a field) was previously established in the case where all residue fields of $R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic $\neq 2$ to the case where the ring has at least one residue field which is finite.