The Artin-springer Theorem for Quadratic Forms over Semi-local Rings with Finite Residue Fields
نویسنده
چکیده
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 étale 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 6= 2 to the case where the ring has at least one residue field which is finite.
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