Abstract
We have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask which integers can be represented in the form x 2 + 2y 2 or, more generally, in the form ax 2 + 2bxy + cy 2, where a, b, c are given integers. The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange’s theorem that every positive integer is a sum of four squares, was continued during the 19th century by Dirichlet, Hermite, H.J.S. Smith, Minkowski and others. In the 20th century Hasse and Siegel made notable contributions. With Hasse’s work especially it became apparent that the theory is more perspicuous if one allows the variables to be rational numbers, rather than integers. This opened the way to the study of quadratic forms over arbitrary fields, with pioneering contributions by Witt (1937) and Pfister (1965–67).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Selected References
E. Artin, Geometric algebra, reprinted, Wiley, New York, 1988. [Original edition, 1957]
T. Beth, D. Jungnickel and H. Lenz, Design theory, 2nd ed., 2 vols., Cambridge University Press, 1999.
A. Borel, Values of indefinite quadratic forms at integral points and flows on spaces of lattices, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 184–204.
J.W.S. Cassels, Rational quadratic forms, Academic Press, London, 1978.
J.W.S. Cassels and A. Fröhlich (ed.), Algebraic number theory, Academic Press, London, 1967.
J.H. Conway, Invariants for quadratic forms, J. Number Theory 5 (1973), 390–404.
S.G. Dani and G.A. Margulis, Values of quadratic forms at integral points: an elementary approach, Enseign. Math. 36 (1990), 143–174.
J. Dieudonné, La géométrie des groupes classiques, 2nd ed., Springer-Verlag, Berlin, 1963.
A. Fröhlich, Quadratic forms ‘à la’ local theory, Proc. Camb. Phil. Soc. 63 (1967), 579–586.
D. Garbanati, Class field theory summarized, Rocky Mountain J. Math. 11 (1981), 195–225.
B. Green, F. Pop and P. Roquette, On Rumely's local-global principle, Jahresber. Deutsch. Math.–Verein. 97 (1995), 43–74.
I. Gusić, Weak Hasse principle for cubic forms, Glas. Mat. Ser. III 30 (1995), 17–24.
H. Hasse, Mathematische Abhandlungen (ed. H.W. Leopoldt and P. Roquette), Band I, de Gruyter, Berlin, 1975.
J.S. Hsia, On the Hasse principle for quadratic forms, Proc. Amer. Math. Soc. 39 (1973), 468–470.
N. Jacobson, Basic Algebra I, 2nd ed., Freeman, New York, 1985.
Y. Kitaoka, Arithmetic of quadratic forms, Cambridge University Press, 1993.
C.W.H. Lam, The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), 305–318.
T.Y. Lam, The algebraic theory of quadratic forms, revised 2nd printing, Benjamin, Reading, Mass., 1980.
D.W. Lewis, The Merkuryev–Suslin theorem, Irish Math. Soc. Newsletter 11 (1984), 29–37.
J. Milnor and D. Husemoller, Symmetric bilinear forms, Springer-Verlag, Berlin, 1973.
J. Neukirch, Class field theory, Springer-Verlag, Berlin, 1986.
O.T. O'Meara, Introduction to quadratic forms, corrected reprint, Springer-Verlag, New York, 1999. [Original edition, 1963]
A. Pfister, Hilbert's seventeenth problem and related problems on definite forms, Mathematical developments arising from Hilbert problems (ed. F.E. Browder), pp. 483–489, Proc. Symp. Pure Math. 28, Part 2, Amer. Math. Soc., Providence, Rhode Island, 1976.
A. Pfister, Quadratic forms with applications to algebraic geometry and topology, Cambridge University Press, 1995.
A.R. Rajwade, Squares, Cambridge University Press, 1993.
M. Ratner, Interactions between ergodic theory, Lie groups, and number theory. Proceedings of the International Congress of Mathematicians: Zürich 1994, pp. 157–182, Birkhäuser, Basel, 1995.
W. Rudin, Sums of squares of polynomials, Amer. Math. Monthly 107 (2000), 813–821.
W. Scharlau, Quadratic and Hermitian forms, Springer-Verlag, Berlin, 1985.
J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973.
W.C. Waterhouse, Pairs of quadratic forms, Invent. Math. 37 (1976), 157–164.
K.S. Williams, On the size of a solution of Legendre's equation, Utilitas Math. 34 (1988), 65–72.
E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31–44.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Coppel, W.A. (2009). The Arithmetic of Quadratic Forms. In: Number Theory. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89486-7_7
Download citation
DOI: https://doi.org/10.1007/978-0-387-89486-7_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-89485-0
Online ISBN: 978-0-387-89486-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)