4. Conclusion
There are many ways to measure the reception — and success — of a text, or, more precisely, the ideas of a text. We have elaborated here, in part, on Martin Kneser’s personal remarks and contributions at Oberwolfach, in June 2001. That Gauss’s question on the composition of forms could lead to the construction of a complex, yet stunningly elegant, algebraic theory of composition for binary quadratic modules over an arbitrary commutative ring with unity certainly testifies — again — to the breadth and depth of Gauss’s original ideas. Hurwitz’s private thoughts on Gauss’s composition of forms at the close of the nineteenth century, Bhargava’s more contemporary results on the arithmetic of number fields and Kneser’s extension of the theory via Clifford algebras provide further evidence of the lasting impact of the Disquisitiones Arithmeticae.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Belabas, Karim. 2005. Paramétrisation de structures algébriques et densité de discriminants (d’après Bhargava). In Séminaire Bourbaki. Vol. 2003–2004, exposé 935, pp. 267–299. Astérisque 299. Paris: Société mathématique de France.
Bhargava, Manjul. 2004. Higher composition laws. I. A new view on Gauss composition, and quadratic generalizations. Annals of Maths 159, 217–250. II. On cubic analogues of Gauss composition. Annals of Maths 159, 865–886. III. The parametrization of quartic rings. Annals of Maths 159, 1329–1360.
____. 2005. The density of discriminants of quartic rings and fields. Annals of Maths 162, 1031–1062.
Brandt, Heinrich. 1913. Zur Komposition der quaternären quadratischen Formen. Journal für die reine und angewandte Mathematik 143, 106–129.
____. 1924. Der Kompositionsbegriff bei den quaternären quadratischen Formen. Mathematische Annalen 91, 300–315.
Dickson, Leonard E. 1923. History of the Theory of Numbers, vol. III. Washington D.C.: Carnegie Institution of Washington.
Dulin, Bill J., Butts, Hubert S. 1972. Composition of binary quadratic forms over integral domains. Acta Arithmetica 20, 223–251.
Gauss, Carl Friedrich. 1866. Werke, ed. Königliche Gesellschaft der Wissenschaften zu Göttingen, vol. III. Göttingen: Universitäts-Druckerei.
Hahn, Alexander, O’Meara, O. Timothy. 1989. The Classical Groups and K-Theory. Grundlehren der mathematischen Wissenschaften 291. Berlin, Heidelberg: Springer.
Hurwitz, Adolf. 1898. Über die Komposition der quadratischen Formen von beliebig vielen Veränderlichen. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 309–316. Repr. Mathematische Werke, vol. II, pp. 565–571. Basel, Stuttgart: Birkhäuser, 1963.
Jacobson, Nathan. 1958. Composition algebras and their automorphisms. Rendiconti del Circolo Matematico di Palermo 2nd ser. 7, 55–80.
Kaplansky, Irving. 1968. Composition of binary quadratic forms. Studia Mathematica 31, 523–530.
Kneser, Martin. 1982. Composition of binary quadratic forms. Journal of Number Theory 15, 406–413.
____. 1987. Komposition quadratischer Formen. Wissenschaftliche Beiträge der Martin-Luther-Universität Halle-Wittenberg. 1987/33 (M48), 161–173.
____. 2002. Quadratische Formen. Berlin, Heidelberg, New York: Springer.
Kneser, Martin, Knus, Max-Albert, Ojanguren, Manuel, Parimala, Raman, Sridharan, Raja. 1986. Composition of quaternary quadratic forms. Compositio Mathematica 60, 133–150.
Kummer, Ernst Eduard. 1975. Collected Papers, ed. A. Weil. 2 vols. Berlin: Springer.
Neumann, Olaf. 1979. Bemerkungen aus heutiger Sicht über Gauß’ Beiträge zur Zahlentheorie, Algebra und Funktionentheorie. NTM-Schriftenreihe 16:2, 22–39.
____. 1980. Zur Genesis der algebraischen Zahlentheorie. Bemerkungen aus heutiger Sicht über Gauss’ Beiträge zu Zahlentheorie, Algebra und Funktionentheorie. 2. und 3. Teil. NTM-Schriftenreihe 17:1, 32–48; 17:2, 38–58.
Smith, Henry J. S. 1862. Report on the Theory of Numbers, part IV. Report of the British Association for the Advancement of Science for 1862, 503–526. Repr. in Collected Mathematical Papers, ed. J. W. L. Glaisher, vol. I, pp. 229–262. Oxford: Clarendon, 1894.
Springer, Tonny Albert, Veldkamp, Ferdinand D. 2000. Octonions, Jordan Algebras and Exceptional Groups. Berlin, Heidelberg, New York: Springer.
Towber, James. 1980. Composition of oriented binary quadratic form-classes over commutative rings. Advances in Mathematics 36, 1–107.
Waterhouse, William C. 1984. Composition of norm-type forms. Journal für die reine und angewandte Mathematik 353, 85–97.
Weil, André. 1984. Number Theory. An Approach through History from Hammurapi to Legendre. Boston, Bastel, Stuttgart: Birkhäuser.
____. 1986. Gauss et la composition des formes quadratiques binaires. In Aspects of Mathematics and its Applications, ed. J.A. Barroso, pp. 895–912. North-Holland Mathematical Library 34. Amsterdam: North-Holland, Elsevier.
Witt, Ernst. 1937. Theorie der quadratischen Formen in beliebigen Körpern. Journal für die reine und angewandte Mathematik 176, 31–44.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fenster, D.D., Schwermer, J. (2007). Composition of Quadratic Forms: An Algebraic Perspective. In: Goldstein, C., Schappacher, N., Schwermer, J. (eds) The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34720-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-34720-0_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20441-1
Online ISBN: 978-3-540-34720-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)