| Revision as of 22:42, 2 March 2011 edit212.183.140.15 (talk) link← Previous edit | Revision as of 22:44, 2 March 2011 edit undo212.183.140.15 (talk) →References: Milnor and HusemollerNext edit → | ||
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| *{{Citation | editor1-last=Iyanaga | editor1-first=Shôkichi | editor2-last=Kawada | editor2-first=Yukiyosi | title=Encyclopedic dictionary of mathematics, Volumes I, II| origyear=1977 | publisher=] | edition=1st | series=Translated from the 2nd Japanese edition, paperback version of the 1977 edition | isbn=978-0-262-59010-5 | id={{MR|591028}} | year=1980}} | *{{Citation | editor1-last=Iyanaga | editor1-first=Shôkichi | editor2-last=Kawada | editor2-first=Yukiyosi | title=Encyclopedic dictionary of mathematics, Volumes I, II| origyear=1977 | publisher=] | edition=1st | series=Translated from the 2nd Japanese edition, paperback version of the 1977 edition | isbn=978-0-262-59010-5 | id={{MR|591028}} | year=1980}} | ||
| *{{Citation | last1=Lam | first1=T. Y. | title=Introduction to quadratic forms over fields | url=http://books.google.com/books?id=YvyOLDeOYQgC | publisher=] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-1095-8 | id={{MR|2104929}} | year=2005 | volume=67|chapter=Chapter VIII section 4: Pythagorean fields|page=255-264}} | *{{Citation | last1=Lam | first1=T. Y. | title=Introduction to quadratic forms over fields | url=http://books.google.com/books?id=YvyOLDeOYQgC | publisher=] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-1095-8 | id={{MR|2104929}} | year=2005 | volume=67|chapter=Chapter VIII section 4: Pythagorean fields|page=255-264}} | ||
| *{{citation | first1=J. | last1=Milnor | first2=D. | last2=Husemoller | title=Symmetric bilinear forms | publisher=Springer | year=1973 | isbn=3-540-06009-X | page=71 }} | |||
| ] | ] | ||
Revision as of 22:44, 2 March 2011
In algebra, a Pythagorean field is a field in which every sum of two squares is a square. A Pythagorean extension of a field F is an extension obtained by adjoining an element √1 + λ for some λ in F. So a Pythagorean field is one closed under taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism, called its Pythagorean closure.
Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry (Ito 1980, 163 C) harv error: no target: CITEREFIto1980 (help). The analytic geometry given by F for F a Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels. (In general this geometry need not satisfy all Hilbert's axioms unless the field F has extra properties: for example, if the field is also ordered then the geometry will satisfy Hilbert's ordering axioms, and if the field is also complete the geometry will satisfy Hilbert's completeness axiom.)
The Pythagorean closure of a non-archimedean ordered field, such as the Pythagorean closure of the field of rational functions Q(t) in one variable over the rational numbers Q, can be used to construct non-archimedean geometries that satisfy many of Hilbert's axioms but not his axiom of completeness (Ito 1980, 163 D) harv error: no target: CITEREFIto1980 (help). Dehn used such a field to construct a non-Legendrian geometry and a semi-Euclidean geometry in which there are many lines though a point not intersecting a given line.
The Witt ring of a Pythagorean field is of order 2 if the field is not formally real, and torsion-free otherwise.
See also
References
- Elman, Richard; Lam, T. Y. (1972), "Quadratic forms over formally real fields and pythagorean fields", American Journal of Mathematics, 94: 1155–1194, ISSN 0002-9327, MR0314878
- Iyanaga, Shôkichi; Kawada, Yukiyosi, eds. (1980) , Encyclopedic dictionary of mathematics, Volumes I, II, Translated from the 2nd Japanese edition, paperback version of the 1977 edition (1st ed.), MIT Press, ISBN 978-0-262-59010-5, MR591028
- Lam, T. Y. (2005), "Chapter VIII section 4: Pythagorean fields", Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, Providence, R.I.: American Mathematical Society, p. 255-264, ISBN 978-0-8218-1095-8, MR2104929
- Milnor, J.; Husemoller, D. (1973), Symmetric bilinear forms, Springer, p. 71, ISBN 3-540-06009-X