Pythagoras numbers of fields

Author:
Detlev W. Hoffmann

Journal:
J. Amer. Math. Soc. **12** (1999), 839-848

MSC (1991):
Primary 11E04, 11E10, 11E25, 12D15

Published electronically:
April 13, 1999

MathSciNet review:
1670858

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Abstract: A field of characteristic is said to have finite Pythagoras number if there exists an integer such that each nonzero sum of squares in can be written as a sum of squares, in which case the Pythagoras number of is defined to be the least such integer. As a consequence of Pfister's results on the level of fields, of a nonformally real field is always of the form or , and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form , , and can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer there exists a formally real field with . As a refinement, we will show that if and are integers such that , then there exists a uniquely ordered field with and (resp. ), where (resp. ) denotes the supremum of the dimensions of anisotropic forms over which are torsion in the Witt ring of (resp. which are indefinite with respect to each ordering on ).

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Additional Information

**Detlev W. Hoffmann**

Affiliation:
Equipe de Mathématiques de Besançon, UMR 6623 du CNRS, Université de Franche-Comté, 16, Route de Gray, F-25030 Besançon Cedex, France

Email:
detlev@math.univ-fcomte.fr

DOI:
http://dx.doi.org/10.1090/S0894-0347-99-00301-X

Keywords:
Quadratic forms,
sums of squares,
formally real fields,
Pythagoras number,
$u$-invariant,
Hasse number

Received by editor(s):
July 31, 1998

Received by editor(s) in revised form:
February 12, 1999

Published electronically:
April 13, 1999

Article copyright:
© Copyright 1999
American Mathematical Society