Abstract
Disquisitiones Arithmeticae was published in Leipzig, then the center of the German book trade. This was in the summer of 1801, nearly three years after Gauss had moved back to Brunswick. The following review will not be very ambitious, and we shall limit ourselves to a summary of Gauss’s work, rather than try to evaluate it and its role in the development of number theory.
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Notes
To use indices means to make use of the representation of the cyclic group of the primitive residue classes by a particular generator.
Gauss’s result contains Wilson’s theorem as a special case. It is not the first proof of the theorem.
This is not Gauss’s but Legendre’s formalism. Gauss never used it in his publi-cations and, though he was certainly familiar with it, never clearly referred to it. When he pointed out, in §76 of Disqu. Arithm., that theorems should follow from notions and not from notations, he did this in connection with his proof of Wilson’s theorem and in reference to a remark of Waring. Gauss used Legendre’s formalism in several only posthumously published fragments.
Gauss’s other proofs are similarly elementary but not so direct.
Kronecker called this proof a touchstone (Prüfstein) of Gauss’s genius.
The gap in Legendre’s proof consists in the assumption, only proved by Dirichlet considerably later, that any arithmetic progression ax + b, (a, b) = 1, contains an infinite number of primes.
Legendre’s binary quadratic forms were defined as \( a{x^2} + bxy + c{y^2}. \) It is computationally advantageous to assume an even coefficient for the term in the middle.
Gauss calls this function the determinant.
One finds more about this question in the literature, eg., in [Edwards].
It would have been difficult for Gauss to confine himself to the theory of binary quadratic forms. Only much later, more than 100 years after the publication of Disqu. Arithm., an elementary proof of the substance of §287, the real object for the disgression into ternary forms, was found.
In the course of its several editions, Dedekind added more and more appendices to the book and specifically introduced ideal theory into arithmetic. The five Chapters of [Dirichlet-Dedekind] cover the following topics: divisibility, congruence, quadratic residues, quadratic forms, and the class number. The four supplements are devoted to the division of the circle, Pell’s equation, the composition of forms, and the theory of algebraic integers.
K. Hensel wrote the following in his introduction to Kronecker’s Vorlesungen über Zahlentheorie: “… Gauss hat die Arithmetik zum Range einer Wissenschaft erhoben, aber erst Dirichlet gab ihr, wie schon Kronecker mit Recht hervorhob, wirklich eigentliche Methoden, indem er zeigte, dass und wie man ganze Klassen arithmetischer Probleme entweder lösen, oder wenigstens die arithmetische Schwierigkeit auf eine analytische reduzieren kann. Die Methoden Dirichlet’s beruhen wesentlich auf der Einführung des Grenzbegriffes in die Arithmetik…”
… Was da … [Disqu. Arithm. §356]… steht ist streng dort bewiesen, aber was fehlt, nämlich die Bestimmung des Wurzelzeichens, ist es gerade, was mich immer gequält hat. Dieser Mangel hat mir alles Übrige, was ich fand, verleidet, und seit vier Jahren wird selten eine Woche hingegangen sein, wo ich nicht einen oder den anderen vergeblichen Versuch, diesen Knoten zu lösen, gemacht hätte besonders lebhaft nun auch wieder in der letzten Zeit. Aber alles Brüten, alles Suchen ist umsonst gewesen, traurig habe ich jedesmal wieder die Feder niederlegen müssen. Endlich vor ein paar Tagen ist es mir gelungen—aber nicht meinem mühsamen Suchen sondern bloss durch die Gnade Gottes möchte ich sagen. Wie der Blitz einschlägt, hat sich das Räthsel gelöst; ich selbst wäre nicht im Stande, den leitenden Faden zwischen dem, was ich vorher wusste, dem, womit ich die letzten Versuche gemacht hatte—und dem, wodurch es gelang nachzuweisen….
See, eg., the letters to Bessel of June 28,1820 (#45), and of March 12,1826 (#58), or the letter to Dirichlet of Nov. 2,1838.
The second volume of Kronecker’s collected papers contains a historical review of the law of quadratic reciprocity. One finds relatively complimentary remarks about Legendre’s incomplete proof in Gauss’s appendix of Disqu. Arithm. and in his “Theorematis arithmetici demonstratio nova” of 1808.
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© 1981 Springer-Verlag New York Inc.
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Bühler, W.K. (1981). The Number-Theoretical Work. In: Gauss. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-49207-5_6
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