Disquisitiones Arithmeticae

from Wikipedia, the free encyclopedia
Title page of the first edition

The Disquisitiones Arithmeticae (Latin for number-theoretical investigations ) are a textbook on number theory ("Higher Arithmetic" in Gauss' words) that the German mathematician Carl Friedrich Gauß wrote in 1798 at the age of 21 and that was published on September 29, 1801 in Leipzig. In this work, Gauss created, in the words of Felix Klein, "in the true sense of the word modern number theory and has determined all of the following developments to this day". In it he presents earlier results by Pierre de Fermat , Leonhard Euler , Joseph Louis Lagrange and Adrien-Marie Legendre (the authors who Gauß himself explicitly mentions in the foreword next to Diophantus ) as well as numerous own discoveries and developments in a systematic way. The book is as one of the last great mathematical works written in Latin. Elementary number theory (Chapters 1 to 3) are dealt with and the foundations of algebraic number theory are laid. The book is written in the classic sentence-proof-corollary style, contains no motivation for the lines of evidence taken and carefully hides the way in which Gauss came to his discoveries. Gauß's work only became accessible to other mathematical circles through the lectures by Peter Gustav Lejeune Dirichlet .

The delay in printing, which began in 1798, was caused, among other things, by problems with the book printers who had to set the difficult work. Nevertheless, corrective pages had to be inserted again in the original. The first four chapters date from 1796 and were essentially in their final form at the end of 1797, when Gauss was still in Göttingen. The first version of the central chapter 5 dates from the summer of 1796, but was revised several times up to the beginning of 1800. From autumn 1798 Gauß was back in Braunschweig, where he lived until 1807.

The dedication to his patron, the Duke of Braunschweig , is dated July 1801. The Duke made the printing possible.

content

  • Chapter 1 (five pages) deals with congruence arithmetic (modular arithmetic) and divisibility rules.
  • Chapter 2 (24 pages) provides the unambiguous prime factorization and the solution of linear equations in modular arithmetic (called "mod n" for short).
  • Chapter 3 (35 pages) deals with powers mod n including the concept of the primitive root and its index (the analog of the logarithm in modular arithmetic). Here you will find the “ little Fermatsche sentence ”, Wilson's theorem and criteria for quadratic remainders.
  • Chapter 4 (47 pages) deals with his "fundamental theorem" of arithmetic, the quadratic reciprocity law , that is, the question of solving quadratic equations in congruence arithmetic. The proof is cumbersome due to the many case distinctions, but is kept "elementary" and is already announced in his diary of 1796. Peter Gustav Lejeune Dirichlet simplified the proof in 1857 using the Jacobi symbol . Gauss does not use Legendre's notation for the Legendre symbol, but rather aRb if a is a quadratic residue of b and aNb if not. The quadratic reciprocity law was the starting point of Gauss's number theoretic work, as he wrote in his foreword.
  • Chapter 5 (almost half of the book with 260 pages) deals with the number theory of binary quadratic forms (in two variables), which Lagrange already dealt with. Equivalence classes of square shapes are introduced and reduced shapes belonging to a class as well as the numbers that can be expressed by shapes of a certain class are characterized. To this end, he defines the order, gender and character of a class. His theory of the composition of square shapes forms the climax.

Paragraph 262 contains a new proof of the quadratic reciprocity law from the theory of quadratic forms, for which Gauss provided several further proofs in the course of his life. This proof is also announced in his diary from 1796. There is also a theory of ternary quadratic forms (in three variables). Paragraph 303 contains his calculations on the class numbers of imaginary square number fields. In particular, Gauss lists all such number fields with 1, 2 and 3 classes. He lists nine imaginary square number fields especially for class number 1 and assumes that these are all: Numbers of the form   ( whole ) with This is the starting point for investigations into the “class number problem”, which in the case of class number 1 by Kurt Heegner , Harold Stark , Alan Baker was solved and generally came to a certain conclusion in the 1980s by Don Zagier and Benedict Gross . Paragraph 293 gives solutions to Fermat's polygonal number problem for squares (which Lagrange already solved) and cubes. Gauss sums appear for the first time in paragraph 356 . A sentence in paragraph 358 was later recognized by André Weil as a special case of the Riemann hypothesis for curves over finite fields (see Weil conjectures ). For another elliptic curve, Gauss put up a sentence equivalent to the Riemann hypothesis in the last entry in his diary (proven by Gustav Herglotz 1921).

  • Chapter 6 deals with, among others, continued fractions . There are also two different primality tests to be found here .
  • Chapter 7 deals with the doctrine of the division of circles. Here is the proof that a corner can be constructed with a compass and ruler if it is a Fermat prime number (explicitly for the 17 corner). But it gives no proof of the impossibility of the construction for other numbers (this was done by Pierre Wantzel ). He also suggests a generalization on the division of the lemniscate .

Many of the sentences are already in Legendre's number theory, which emerged almost at the same time, but they were found independently by Gauss, as he only got to know Legendre's book when a large part of his Disquisitiones were already at the printer (as Gauss in his preface). There was also a resentment of Legendre, who saw himself insufficiently appreciated by Gauss and complained to him about it. Legendre's book was later completely overshadowed by Gauss' Disquisitiones. Gauss planned to continue the Disquisitiones, but it never came about. For example, material on this was published in his treatises on biquadratic residues (1825, 1831), in which he introduced Gaussian numbers . An “eighth chapter” of the Disquisitiones was discovered in the estate ( Analysis Residuorum ) and published in the second volume of the complete edition. According to Gauss in the preface to the Disquisitiones , in which he also refers several times to this eighth chapter, it should deal with generally indeterminate equations in modular arithmetic.

Many profound remarks by Gauss (such as the one on the lemniscate , the starting point of the theory of complex multiplication in algebraic number theory) stimulated mathematicians such as Augustin-Louis Cauchy (who completely solved Fermat's polygonal number problem in 1815), Gotthold Eisenstein , Carl Gustav Jacobi , Ernst Eduard Kummer , Dirichlet (who always had a copy of the Disquisitiones close at hand at his desk), Charles Hermite , Hermann Minkowski , David Hilbert and even André Weil for further investigations. Another example is the expansion of the laws of composition of square forms to include higher laws of composition by Manjul Bhargava from 2004.

expenditure

  • The original edition was published by Gerhard Fleischer, Lipsiae (Leipzig) 1801 (668 pages, octave format). A first reprint appeared as the first volume of the complete edition of Carl Friedrich Gauß: Works. Volume 1 , Dieterich, Göttingen 1863, its second reprint 1870 (in the Internet archive: [2] ), published by the Royal Society of Sciences in Göttingen by Ernst Schering . A facsimile edition was published in Brussels in 1968 (Culture et civilization), a reprint in 2006 by Olms in Hildesheim, Ed. Jochen Brüning, with a foreword by Norbert Schappacher , ISBN 3-487-12845-4 .
  • Carl Friedrich Gauss' studies on higher arithmetic , edited by Hermann Maser, Julius Springer, Berlin 1889 (German translation; in the Internet archive: [3] ), also studies on higher arithmetic . AMS Chelsea Publications 2006, 695 pages, ISBN 0-8218-4213-7 (together with other works by Gauß); the book was reprinted by Verlag Kessel in 2009: ISBN 978-3-941300-09-5
  • Disquisitiones arithmeticae , Yale University Press, 1966, reprint Springer-Verlag, New York Heidelberg 1986, ISBN 0-387-96254-9 (English translation by Arthur A. Clarke, revised in 1986 by William C. Waterhouse)
  • Recherches arithmétiques , Courcier, Paris 1807, reprinted by Jacques Gabay, Paris 1989 (French translation by A.-C.-M. Poullet-Deslisle)
  • Gausu Seisuron . Asakura Publishing Co., Ltd., Tokyo, Japan 1995, ISBN 4-254-11457-5 ( online - Japanese: ガ ウ ス 整数 論 . Translated by Masahito Takase).
  • there are also Russian (Demjanov translation, Moscow 1959), Spanish (1995) and Catalan (1996) editions.

Secondary literature

Web links

swell

  1. ^ Felix Klein : Lectures on the Development of Mathematics in the 19th Century , Julius Springer, Berlin 1926, p. 26
  2. ↑ Klein, loc. Cit., P. 27
  3. ↑ 1837 introduced by Jacobi but implicitly in the book by Gauß. David A. Cox , Primes of the form , Wiley, 1989, p. 64
  4. ↑ A total of six essentially different, Gauss himself distinguishes eight. This proof of the theory of quadratic forms is presented in Daniel Flath, Introduction to number theory, Wiley 1989, p. 163. Flath also represents the first proof and further proofs of Gauss's theorem.
  5. ^ David A. Cox, Primes of the form , Wiley, 1989, p. 86
  6. ↑ Treated in Kenneth Ireland, Michael Rosen: A classical introduction to modern number theory, Springer, 1990, p. 166
  7. ^ Frans Oort : The last entry, Notices ICCM, July 2016, pdf
  8. ↑ already announced in the intelligence paper of the Allgemeine Literaturzeitung, Jena 1796
  9. ^ Carl Friedrich Gauß: Works. Volume 2 , Dieterich, Göttingen 1863, pp. 212-240 (at Google Books: [1] ). At the same time there are preliminary drafts for some of the published chapters in the manuscript. The corresponding sections have been omitted from Volume 2 of the Complete Edition.