Composition of Quadratic Forms: An Algebraic Perspective

Della D. Fenster () and Joachim Schwermer ()
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Della D. Fenster: University of Richmond, Department of Mathematics and Computer Science
Joachim Schwermer: Universität Wien, Fakultät für Mathematik

Chapter II.3 in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, 2007, pp 145-158 from Springer

Abstract: 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.

Keywords: Quadratic Form; Clifford Algebra; Great Common Divisor; Composition Algebra; Binary Quadratic Form (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/978-3-540-34720-0_5

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