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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The reduced Witt ring of a formally real field


Author: Ron Brown
Journal: Trans. Amer. Math. Soc. 230 (1977), 257-292
MSC: Primary 12D15; Secondary 15A63
MathSciNet review: 0472781
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Abstract: The reduced Witt rings of certain formally real fields are computed here in terms of some basic arithmetic invariants of the fields. For some fields, including the rational function field in one variable over the rational numbers and the rational function field in two variables over the real numbers, this is done by computing the image of the total signature map on the Witt ring. For a wider class of fields, including all those with only finitely many square classes, it is done by computing the Witt rings of certain ultracompletions of the field and representing the reduced Witt ring as an appropriate subdirect product of the Witt rings of the ultracompletions. The reduced Witt rings of a still wider class of fields, including for example the fields of transcendence degree one and the rational function field in three variables over the real numbers, are computed similarly, except that the description of the subdirect product no longer involves only local conditions.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0472781-4
PII: S 0002-9947(1977)0472781-4
Keywords: Witt ring, reduced Witt ring, quadratic form, formally real field, ordering, real place, valuation, total signature map, ultracompletion, SAP field, superpythagorean field
Article copyright: © Copyright 1977 American Mathematical Society