Pythagoras numbers of fields
Author:
Detlev W. Hoffmann
Journal:
J. Amer. Math. Soc. 12 (1999), 839848
MSC (1991):
Primary 11E04, 11E10, 11E25, 12D15
Published electronically:
April 13, 1999
MathSciNet review:
1670858
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 ).
 [A]
E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Semin. Hamburg. Univ. 5 (1927) 100115.
 [CEP]
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 0292781
(45 #1863)
 [EL]
Richard
Elman and T.
Y. Lam, Pfister forms and 𝐾theory of fields, J.
Algebra 23 (1972), 181–213. MR 0302739
(46 #1882)
 [ELP]
Richard
Elman, Tsit
Yuen Lam, and Alexander
Prestel, On some Hasse principles over formally real fields,
Math. Z. 134 (1973), 291–301. MR 0330045
(48 #8384)
 [ELW]
R.
Elman, T.
Y. Lam, and A.
R. Wadsworth, Orderings under field extensions, J. Reine
Angew. Math. 306 (1979), 7–27. MR 524644
(80e:12029)
 [EP]
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
(86f:11030), http://dx.doi.org/10.2307/2374279
 [H1]
Detlev
W. Hoffmann, Isotropy of quadratic forms over the function field of
a quadric, Math. Z. 220 (1995), no. 3,
461–476. MR 1362256
(96k:11041), http://dx.doi.org/10.1007/BF02572626
 [H2]
Detlev
W. Hoffmann, Twisted Pfister forms, Doc. Math.
1 (1996), No. 03, 67–102 (electronic). MR 1386048
(97d:11065)
 [H3]
Detlev
W. Hoffmann, On Elman and Lam’s filtration of the
𝑢invariant, J. Reine Angew. Math. 495
(1998), 175–186. MR 1603861
(99d:11037), http://dx.doi.org/10.1515/crll.1998.017
 [Hor]
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
(92f:11055), http://dx.doi.org/10.1016/00193577(91)900426
 [I]
O.T. Izhboldin, On the isotropy of quadratic forms over the function field of a quadric, Algebra i Analiz. 10 (1998), 3257. (Russian). English transl. to appear in St. Petersburg Math. J. 10 (1999). CMP 98:12
 [K]
Manfred
Knebusch, Generic splitting of quadratic forms. II, Proc.
London Math. Soc. (3) 34 (1977), no. 1, 1–31.
MR
0427345 (55 #379)
 [L1]
T.
Y. Lam, The algebraic theory of quadratic forms, W. A.
Benjamin, Inc., Reading, Mass., 1973. Mathematics Lecture Note Series. MR 0396410
(53 #277)
 [L2]
, Some consequences of Merkurjev's work on function fields, Preprint 1989.
 [M]
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. USSRIzv.
38 (1992), no. 1, 215–221. MR 1130036
(93b:16025)
 [Pe]
Meinhard
Peters, Summen von Quadraten in Zahlringen, J. Reine Angew.
Math. 268/269 (1974), 318–323 (German). Collection
of articles dedicated to Helmut Hasse on his seventyfifth birthday, II. MR 0352063
(50 #4551)
 [Pf]
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
(97c:11046)
 [Pr]
Alexander
Prestel, Remarks on the Pythagoras and Hasse number of real
fields, J. Reine Angew. Math. 303/304 (1978),
284–294. MR
514686 (80d:12020), http://dx.doi.org/10.1515/crll.1978.303304.284
 [Sc]
Rudolf
Scharlau, On the Pythagoras number of orders in totally real number
fields, J. Reine Angew. Math. 316 (1980),
208–210. MR
581331 (81g:10036), http://dx.doi.org/10.1515/crll.1980.316.208
 [S]
Winfried
Scharlau, Quadratic and Hermitian forms, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 270, SpringerVerlag, Berlin, 1985. MR 770063
(86k:11022)
 [A]
 E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Semin. Hamburg. Univ. 5 (1927) 100115.
 [CEP]
 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) 125149. MR 45:1863
 [EL]
 R. Elman, T.Y. Lam, Pfister forms and theory of fields, J. Algebra 23 (1972) 181213. MR 46:1882
 [ELP]
 R. Elman, T.Y. Lam, and A. Prestel, On some Hasse principles over formally real fields. Math. Z. 134 (1973) 291301. MR 48:8384
 [ELW]
 R. Elman, T.Y. Lam, and A.R. Wadsworth, Orderings under field extensions, J. Reine Angew. Math. 306 (1979) 727. MR 80e:12029
 [EP]
 R. Elman and A. Prestel, Reduced stability of the Witt ring of a field and its Pythagorean closure, Amer. J. Math. 106 (1983) 12371260. MR 86f:11030
 [H1]
 D.W. Hoffmann, Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995) 461476. MR 96k:11041
 [H2]
 , Twisted Pfister forms, Doc. Math. J. DMV 1 (1996) 67102. MR 97d:11065
 [H3]
 , On Elman and Lam's filtration of the invariant, J. Reine Angew. Math. 495 (1998), 175186. MR 99d:11037
 [Hor]
 E.A.M. Hornix, Formally real fields with prescribed invariants in the theory of quadratic forms, Indag. Math. 2 (1991) 6578. MR 92f:11055
 [I]
 O.T. Izhboldin, On the isotropy of quadratic forms over the function field of a quadric, Algebra i Analiz. 10 (1998), 3257. (Russian). English transl. to appear in St. Petersburg Math. J. 10 (1999). CMP 98:12
 [K]
 M. Knebusch, Generic splitting of quadratic forms. II, Proc. London Math. Soc. (1977) 131. MR 55:379
 [L1]
 T.Y. Lam, The Algebraic Theory of Quadratic Forms, Reading, Massachusetts: Benjamin 1973 (revised printing 1980). MR 53:277
 [L2]
 , Some consequences of Merkurjev's work on function fields, Preprint 1989.
 [M]
 A.S. Merkurjev, Simple algebras and quadratic forms, Izv. Akad. Nauk. SSSR 55 (1991) 218224. (English translation: Math. USSR Izvestiya 38 (1992) 215221.) MR 93b:16025
 [Pe]
 M. Peters, Summe von Quadraten in Zahlringen, J. Reine Angew. Math. 268/269 (1974) 318323. MR 50:4551
 [Pf]
 A. Pfister, Quadratic forms with applications to algebraic geometry and topology, London Math. Soc. Lect. Notes 217, Cambridge University Press 1995. MR 97c:11046
 [Pr]
 A. Prestel, Remarks on the Pythagoras and Hasse number of real fields, J. Reine Angew. Math. 303/304 (1978) 284294. MR 80d:12020
 [Sc]
 R. Scharlau, On the Pythagoras number of orders in totally real number fields, J. Reine Angew. Math. 316 (1980) 208210. MR 81g:10036
 [S]
 W. Scharlau, Quadratic and Hermitian Forms, Grundlehren 270, Berlin, Heidelberg, New York, Tokyo: Springer 1985. MR 86k:11022
Similar Articles
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 FrancheComté, 16, Route de Gray, F25030 Besançon Cedex, France
Email:
detlev@math.univfcomte.fr
DOI:
http://dx.doi.org/10.1090/S089403479900301X
PII:
S 08940347(99)00301X
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
