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

Pythagoras numbers of fields

Author: Detlev W. Hoffmann
Journal: J. Amer. Math. Soc. 12 (1999), 839-848
MSC (1991): Primary 11E04, 11E10, 11E25, 12D15
DOI: https://doi.org/10.1090/S0894-0347-99-00301-X
Published electronically: April 13, 1999
MathSciNet review: 1670858
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Abstract: 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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Keywords: Quadratic forms, sums of squares, formally real fields, Pythagoras number, $u$-invariant, Hasse number