by Markus Rost (4 pages)
C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 4, 189-192.
Let Z be a complete variety over k. We prove the following purity theorem: an element of the function field K of a proper smooth variety X over k, which for any point of X of codimension 1 can be written as a unit times a product of norms from the residue class fields of closed points of ZK, has the same property with respect to any point of X. If X is rational, one may find a global unit with this property.
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In typical applications one takes for Z a homogenous projective
variety. For instance, if Z is a quadric associated with a Pfister
form f, then the group of norms from the residue class fields of
closed points of Z is the set of values of f.
The following text uses this for the case f=sum of 2d
squares.
von Jean-Louis Colliot-Thélène.
Eberhard Becker zum 60. Geburtstag gewidmet.
Math. Z. 249 3, 2005, 541-543.
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