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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Pythagoras numbers of fields
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by Detlev W. Hoffmann PDF
J. Amer. Math. Soc. 12 (1999), 839-848 Request permission


A field $F$ of characteristic $\neq 2$ is said to have finite Pythagoras number if there exists an integer $m\geq 1$ such that each nonzero sum of squares in $F$ can be written as a sum of $\leq m$ squares, in which case the Pythagoras number $p(F)$ of $F$ is defined to be the least such integer. As a consequence of Pfister’s results on the level of fields, $p(F)$ of a nonformally real field $F$ is always of the form $2^n$ or $2^n+1$, and all integers of such type can be realized as Pythagoras numbers of nonformally real fields. Prestel showed that values of the form $2^n$, $2^n+1$, and $\infty$ can always be realized as Pythagoras numbers of formally real fields. We will show that in fact to every integer $n\geq 1$ there exists a formally real field $F$ with $p(F)=n$. As a refinement, we will show that if $n,m\geq 2$ and $k\geq 1$ are integers such that $2m\geq 2^{k}\geq n$, then there exists a uniquely ordered field $F$ with $p(F)=n$ and $u(F)=\tilde {u}(F)=2m$ (resp. $u(F)=\tilde {u}(F)=\infty$), where $u$ (resp. $\tilde {u}$) denotes the supremum of the dimensions of anisotropic forms over $F$ which are torsion in the Witt ring of $F$ (resp. which are indefinite with respect to each ordering on $F$).
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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:
  • Received by editor(s): July 31, 1998
  • Received by editor(s) in revised form: February 12, 1999
  • Published electronically: April 13, 1999
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 839-848
  • MSC (1991): Primary 11E04, 11E10, 11E25, 12D15
  • DOI:
  • MathSciNet review: 1670858