Abstract
Carl Friedrich Gauss established himself as a mathematician at the age of 24 with the publication of his Disquisitiones Arithmeticae, which eclipsed all previous presentations of number theory and became the standard foundation of future research for a century. At its heart is a massive reworking of the theory of quadratic forms. In this chapter we first present an overview of the work before we concentrate on how this book illuminated the many puzzling complexities still left after Lagrange’s account.
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Notes
- 1.
Published by Springer in 2007. In the extracts that follow I have largely removed the original footnotes.
- 2.
Gazette nationale ou Le Moniteur universel 80 (1807), 312.
- 3.
See Merz (1896–1917), vol. I, pp. 181, 181–182 (footnote), 187–188 and 721.
- 4.
See Del Centina and Fiocca (2012, 693).
- 5.
The Opuscula Analytica is a two-volume edition of previously unpublished works by Euler that was published by the Imperial Academy of Sciences in St. Petersburg in 1783 and 1785. Most of the papers are on number theory, including topics related to quadratic reciprocity.
- 6.
This is the origin of the term ‘character’ in the theory of group representations.
References
Brown, E.: The first proof of the quadratic reciprocity law, revisited. Am. Math. Mon. 88, 257–264 (1981)
Del Centina, A., Fiocca, A.: The correspondence between Sophie Germain and Carl Friedrich Gauss. Arch. Hist. Exact Sci. 66, 585–700 (2012)
Merz, J.T.: History of European Thought in the Nineteenth Century, 4 vols. Blackwood, Edinburgh (1896–1914)
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Gray, J. (2018). Gauss’s Disquisitiones Arithmeticae . In: A History of Abstract Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-94773-0_4
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