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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $F$).

- E. Artin,
*Über die Zerlegung definiter Funktionen in Quadrate*, Abh. Math. Semin. Hamburg. Univ.**5**(1927) 100–115. - J. W. S. Cassels, W. J. Ellison, and A. Pfister,
*On sums of squares and on elliptic curves over function fields*, J. Number Theory**3**(1971), 125–149. MR**292781**, DOI https://doi.org/10.1016/0022-314X%2871%2990030-8 - Richard Elman and T. Y. Lam,
*Pfister forms and $K$-theory of fields*, J. Algebra**23**(1972), 181–213. MR**302739**, DOI https://doi.org/10.1016/0021-8693%2872%2990054-3 - Richard Elman, Tsit Yuen Lam, and Alexander Prestel,
*On some Hasse principles over formally real fields*, Math. Z.**134**(1973), 291–301. MR**330045**, DOI https://doi.org/10.1007/BF01214693 - R. Elman, T. Y. Lam, and A. R. Wadsworth,
*Orderings under field extensions*, J. Reine Angew. Math.**306**(1979), 7–27. MR**524644** - Richard Elman and Alexander Prestel,
*Reduced stability of the Witt ring of a field and its Pythagorean closure*, Amer. J. Math.**106**(1984), no. 5, 1237–1260. MR**761585**, DOI https://doi.org/10.2307/2374279 - Detlev W. Hoffmann,
*Isotropy of quadratic forms over the function field of a quadric*, Math. Z.**220**(1995), no. 3, 461–476. MR**1362256**, DOI https://doi.org/10.1007/BF02572626 - Detlev W. Hoffmann,
*Twisted Pfister forms*, Doc. Math.**1**(1996), No. 03, 67–102. MR**1386048** - Detlev W. Hoffmann,
*On Elman and Lam’s filtration of the $u$-invariant*, J. Reine Angew. Math.**495**(1998), 175–186. MR**1603861**, DOI https://doi.org/10.1515/crll.1998.017 - E. A. M. Hornix,
*Formally real fields with prescribed invariants in the theory of quadratic forms*, Indag. Math. (N.S.)**2**(1991), no. 1, 65–78. MR**1104832**, DOI https://doi.org/10.1016/0019-3577%2891%2990042-6 - O.T. Izhboldin,
*On the isotropy of quadratic forms over the function field of a quadric*, Algebra i Analiz.**10**(1998), 32–57. (Russian). English transl. to appear in St. Petersburg Math. J.**10**(1999). - Manfred Knebusch,
*Generic splitting of quadratic forms. II*, Proc. London Math. Soc. (3)**34**(1977), no. 1, 1–31. MR**427345**, DOI https://doi.org/10.1112/plms/s3-34.1.1 - T. Y. Lam,
*The algebraic theory of quadratic forms*, W. A. Benjamin, Inc., Reading, Mass., 1973. Mathematics Lecture Note Series. MR**0396410** - ---,
*Some consequences of Merkurjev’s work on function fields*, Preprint 1989. - A. S. Merkur′ev,
*Simple algebras and quadratic forms*, Izv. Akad. Nauk SSSR Ser. Mat.**55**(1991), no. 1, 218–224 (Russian); English transl., Math. USSR-Izv.**38**(1992), no. 1, 215–221. MR**1130036** - Meinhard Peters,
*Summen von Quadraten in Zahlringen*, J. Reine Angew. Math.**268(269)**(1974), 318–323 (German). MR**352063**, DOI https://doi.org/10.1515/crll.1974.268-269.318 - Albrecht Pfister,
*Quadratic forms with applications to algebraic geometry and topology*, London Mathematical Society Lecture Note Series, vol. 217, Cambridge University Press, Cambridge, 1995. MR**1366652** - Alexander Prestel,
*Remarks on the Pythagoras and Hasse number of real fields*, J. Reine Angew. Math.**303(304)**(1978), 284–294. MR**514686**, DOI https://doi.org/10.1515/crll.1978.303-304.284 - Rudolf Scharlau,
*On the Pythagoras number of orders in totally real number fields*, J. Reine Angew. Math.**316**(1980), 208–210. MR**581331**, DOI https://doi.org/10.1515/crll.1980.316.208 - Winfried Scharlau,
*Quadratic and Hermitian forms*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 270, Springer-Verlag, Berlin, 1985. MR**770063**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
11E04,
11E10,
11E25,
12D15

Retrieve articles in all journals with MSC (1991): 11E04, 11E10, 11E25, 12D15

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

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