Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae (Latin for number theoretical studies ) are a textbook of number theory ( " Higher Arithmetic" in Gauss's words ), which in 1798 wrote the German mathematician Carl Friedrich Gauss, with only 21 years and was released on 29 September 1801 in Leipzig. In this work Gauss created, in the words of Felix Klein " in the true sense of the modern number theory and certain to this day the whole subsequent development ". He puts it earlier results of Pierre de Fermat, Leonhard Euler, Joseph Louis Lagrange and Adrien -Marie Legendre (the authors, the Gauss next to Diophantus himself explicitly mentioned in the preface) as well as numerous dar. own discoveries and developments in a systematic way the book is as one of the last great mathematical works written in Latin. It will be treated as placed the foundations of algebraic number theory both elementary number theory (up to 3, Chapter 1). The book is written in the classical theorem-proof - corollaries style, contains no motivation of the adopted proof directions carefully and hides the way, as Gauss came to his discoveries. Furthermore, mathematical circles accessible the work of Gauss was only by the lectures of Peter Gustav Lejeune Dirichlet.

The delay of printing that began in 1798 was caused by, among other problems with the printers who had to put the difficult work. Nevertheless, in the original inlaid correction pages were required again. The first four chapters still date from 1796 and were the end of 1797, when Gauss was still in Göttingen, essentially in its final form. The first version of the central chapter 5 is from the summer of 1796, but was reworked several times until early 1800. From the autumn of 1798 Gauss was back in Braunschweig, where he lived until 1807.

The dedication to his patron, the Duke of Brunswick is dated July 1801. The Duke had allows printing only.

Content

• Chapter 1 ( five pages ) deals with the congruence arithmetic (modular arithmetic) and divisibility.
• Chapter 2 ( 24 pages) brings the unique prime factorization and the solution of linear equations in modular arithmetic ( briefly called "mod n").
• Chapter 3 ( 35 pages ) deals with powers mod n including the concept of primitive root and the associated index ( the analogue of the logarithm in the modular arithmetic). Here are the "little Fermat theorem ", the set of Wilson and criteria for quadratic residues.
• Chapter 4 (47 pages) to be treated " fundamental theorem " of arithmetic, the quadratic reciprocity law, ie the question of the dissolution of quadratic equations in the Kongruenzarithmetik. The proof is complicated, but kept "elementary " and finds himself already announced in his diary of 1796. The quadratic reciprocity law was indeed the starting point of the number-theoretic work of Gauss, as he wrote in his preface.
• Chapter 5 (with 260 pages, almost half the book ) deals with the binary number theory of quadratic forms (in two variables), the treated already Lagrange. There are equivalence classes of quadratic forms introduced and corresponding reduced forms, as well as the numbers characterizes a class that can be expressed by forms of a particular class. To this end, he defined order, gender and character of a class. The highlight is his theory of composition of quadratic forms.

In paragraph 262 is a new proof of the quadratic reciprocity law from the theory of quadratic forms, supplied several further evidence of the Gaussian in the course of his life. Also this proof is found already announced in his diary of 1796. In addition, a theory of ternary quadratic forms can be found here (in three variables). In paragraph 303 are his calculations about the class numbers of imaginary quadratic number fields. In particular, Gauss are lists for all such number fields with 1, 2 and 3 classes. Especially for the class number 1 he lists nine imaginary quadratic number fields and suggested that this may all: numbers of the form (whole) with This is the starting point for investigations into the " class number problem" that in the case of class number 1 by Kurt Heegner, Harold Stark Alan Baker has been solved and generally by Don Zagier and Benedict Gross found a certain degree in the 1980s. In paragraph 356 for the first time appear on Gauss sums. A sentence of paragraph 358 was recognized later by André Weil as a special case of the Riemann hypothesis for curves over finite fields (see Weil conjectures ).

• Chapter 6 deals with, among others, continued fractions. Here are also two different primality tests.
• Chapter 7 deals with the theory of cyclotomic. Here lies the proof that a -gon by ruler and compass can be constructed if a Fermat prime ( explicitly for the 17-gon ). But he gives no proof of the impossibility of construction for other numbers. He also suggests a generalization to the division of the lemniscate.

Many of the sets are already in almost simultaneously resulting number theory by Legendre, however, they were found independently by Gauss, as he only met Legendre's book, as a large part of his Disquisitiones were already at the printer ( as Gauss in his preface). There was also a detuning of Legendre, who saw himself insufficiently recognized by Gauss and complained to about this. Legendre's book was later completely in the shadow of Gauss ' Disquisitiones. Gauss was planning a sequel to the Disquisitiones, but it never came about. Material for it was published, for example, in his treatises on biquadratic residues (1825, 1831), in which he introduces Gaussian numbers. An "eighth chapter" of Disquisitiones was discovered in the estate (Analysis Residuorum ) and published in the second volume of the complete edition. It should, as Gauss in the preface of Disquisitiones in which he repeatedly refers to this eighth chapter, treat generally indeterminate equations in modular arithmetic.

Many profound observations of Gauss (like the lemniscate, the starting point of the theory of complex multiplication in algebraic number theory ) stimulated mathematicians such as Gotthold Eisenstein, Carl Gustav Jacobi, Ernst Eduard Kummer, Dirichlet (which is a copy of the Disquisitiones always ready at his desk had ), Charles Hermite, Hermann Minkowski, David Hilbert, and even more to further investigations of André Weil.

Expenditure

• The original edition was published by Gerhard Fleischer, Lipsiae (Leipzig) 1801 ( 668 pages, octavo ). A first emphasis appeared the first volume of the complete edition of Carl Friedrich Gauss: works. Volume 1, Dieterich, Göttingen 1863, the second impression 1870 (on the Internet Archive :), published by the Royal Society of Sciences in Göttingen by Ernst Schering. A facsimile edition was published in 1968 in Brussels ( Culture et civilization ), reprinted in 2006 Olms in Hildesheim, eds Jochen Brüning, with a foreword by Norbert Schappacher, ISBN 3-487-12845-4.
• Carl Friedrich Gauss ' investigations of higher arithmetic, publisher Hermann Maser, Julius Springer, Berlin 1889 ( German translation, the Internet Archive :), also studies on higher arithmetic. AMS Chelsea Publications 2006, 695 pages, ISBN 0-8218-4213-7 (along with other works of Gauss ); the book was reissued by the publisher boiler 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 in 1807, reprinting Jacques Gabay, Paris, 1989 ( French translation of A.-C.-M. Poullet - Deslisle )
• Gausu Seisuron. Asakura Publishing Co., Ltd., Tokyo, Japan 1995 ( Original title:ガウス 整数 論, translated by Masahito Takase ). , ISBN 4-254-11457-5 (online).
• There are also Russian ( translation Demjanov, Moscow 1959), Spanish (1995 ) and Catalan (1996 ) editions.

Secondary literature

• Jay Goldman: The Queen of Mathematics: A Historically Motivated Guide to Number Theory. A. K. Peters 1997, ISBN 1-56881-006-7.
• Catherine Goldstein, Norbert Schappacher, Joachim Schwermer (Editor): The Shaping of Arithmetic after CF Gauss 's Disquisitiones Arithmeticae. Springer -Verlag, Berlin, Heidelberg, New York, 2007.
• Harold M. Edwards: Composition of Binary Quadratic Forms and the Foundations of Mathematics ( PDF file, with list of editions and translations of Disquisitiones, 340 kB), in: The Shaping of Arithmetic after CF Gauss 's Disquisitiones Arithmeticae, pp. 129-144
• Walter Kaufmann- Bühler: Gauss: A Biographical Study. Springer -Verlag 1981, Chapter 3, ISBN 0-387-10662-6.
• Uta Untermerzbach An early version of Gauss ' disquisitions Arithmeticae, in JW staves Mathematical Perspectives. Essays on mathematics in its historical development, Academic Press, 1981, pp. 167-178
• Olaf Neumann Carl Friedrich Gauss, Disquisitiones Arithmeticae (1801 ) in Ivor Grattan - Guinness Landmark writings in western mathematics 1640-1940, Elsevier 2005, Chapter 22