Pythagoras numbers of fields

Hoffmann, Detlev¹

¹Equipe de Mathématiques de Besançon, UMR 6623 du CNRS, Université de Franche-Comté, 16, Route de Gray, F-25030 Besançon Cedex, France

Journal of the American Mathematical Society, Tome 12 (1999) no. 3, pp. 839-848

Résumé

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 $F$).

Détail

DOI : 10.1090/S0894-0347-99-00301-X

Bibliographie

[1] Cassels, J. W. S., Ellison, W. J., Pfister, A. On sums of squares and on elliptic curves over function fields J. Number Theory 1971 125 149

[2] Elman, Richard, Lam, T. Y. Pfister forms and 𝐾-theory of fields J. Algebra 1972 181 213

[3] Elman, Richard, Lam, Tsit Yuen, Prestel, Alexander On some Hasse principles over formally real fields Math. Z. 1973 291 301

[4] Elman, R., Lam, T. Y., Wadsworth, A. R. Orderings under field extensions J. Reine Angew. Math. 1979 7 27

[5] Elman, Richard, Prestel, Alexander Reduced stability of the Witt ring of a field and its Pythagorean closure Amer. J. Math. 1984 1237 1260

[6] Hoffmann, Detlev W. Isotropy of quadratic forms over the function field of a quadric Math. Z. 1995 461 476

[7] Hoffmann, Detlev W. Twisted Pfister forms Doc. Math. 1996

[8] Hoffmann, Detlev W. On Elman and Lam’s filtration of the 𝑢-invariant J. Reine Angew. Math. 1998 175 186

[9] Hornix, E. A. M. Formally real fields with prescribed invariants in the theory of quadratic forms Indag. Math. (N.S.) 1991 65 78

[10] Knebusch, Manfred Generic splitting of quadratic forms. II Proc. London Math. Soc. (3) 1977 1 31

[11] Lam, T. Y. The algebraic theory of quadratic forms 1973

[12] Merkur′Ev, A. S. Simple algebras and quadratic forms Izv. Akad. Nauk SSSR Ser. Mat. 1991 218 224

[13] Peters, Meinhard Summen von Quadraten in Zahlringen J. Reine Angew. Math. 1974 318 323

[14] Pfister, Albrecht Quadratic forms with applications to algebraic geometry and topology 1995

[15] Prestel, Alexander Remarks on the Pythagoras and Hasse number of real fields J. Reine Angew. Math. 1978 284 294

[16] Scharlau, Rudolf On the Pythagoras number of orders in totally real number fields J. Reine Angew. Math. 1980 208 210

[17] Scharlau, Winfried Quadratic and Hermitian forms 1985