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Encyclopedia > Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. This article needs to be cleaned up to conform to a higher standard of quality. ... This article is in need of attention from an expert on the subject. ... Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... The Union Jack, flag of the newly formed United Kingdom of Great Britain and Ireland. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ... Joseph Louis Lagrange Joseph Louis Lagrange (January 25, 1736 – April 10, 1813; born Giuseppe Luigi Lagrangia in Turin, Lagrange moved to Paris (1787) and became a French citizen, adopting the French translation of his name, Joseph Louis Lagrange) was an Italian mathematician and astronomer who made important contributions to classical... Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ...

Contents


Scope

The Disquisitiones covers both elementary number theory and parts of the area of mathematics that we now call algebraic number theory. However, Gauss did not explicitly recognise the concept of the group that is central to modern algebra, so he did not use this term. His own title for his subject is Higher Arithmetic. In his Preface to the Disquisitiones Gauss describes the scope of the book as follows: In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ...

The inquiries which this volume will investigate pertain to that part of Mathematics which concerns itself with integers.

Contents

The book is divided into seven sections, which are :-

Section I. Congruent Number in General
Section II. Congruences of the First Degree
Section III. Residues of Powers
Section IV. Congruences of the Second Degree
Section V. Forms and Indeterminate Equations of the Second Degree
Section VI. Various Applications of the Preceding Discussions
Section VII. Equations Defining Section of a Circle

Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and treat it in a systematic way. He was also the first mathematician to realise the importance of the property of unique factorisation (sometimes called the fundamental theorem of arithmetic), which he states and proves explicity. Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you take some number a, multiply it by itself p times and subtract a, the result is divisible by p... In mathematics, Wilsons Theorem states that for a prime number p, (see factorial and modular arithmetic for the notation). ... A primitive root modulo n is a concept from modular arithmetic in number theory. ... This article is about the mathematical concept. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...


From Section IV onwards, much of the work is original. Section IV itself develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary quadratic forms; and Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible i.e. can be constructed with a compass and unmarked straight edge alone. In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... A primality test is an algorithm for determining whether an input number is prime. ... In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ... Look up Polygon in Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ...


Gauss started to write an eighth section on higher order congruences, but he did not complete this, and it was published separately after his death.


The Disquisitiones was one of the last mathematical works to be written in scholarly Latin (an English translation was not published until 1965). Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ... 1965 (MCMLXV) was a common year starting on Friday (link goes to calendar). ...


Importance

Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.


The logical structure of the Disquisitiones (theorem statement followed by proof, followed by corollaries) set a standard for later texts. While recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... Look up Proof on Wiktionary, the free dictionary The word proof can mean: Shit and wanker originally, a test assessing the validity or quality of something. ... A theorem is a statement which can be proven true within some logical framework. ...


The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Kummer, Dirichlet and Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... World map showing Europe Europe is conventionally considered one of the seven continents which, in this case, is more a cultural and political distinction than a physiogeographic one. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ... In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at...


References

  • Carl Friedrich Gauss tr. Arthur A. Clarke: Disquisitiones Aritmeticae, Yale University Press, 1965 ISBN 0300094736
  • Disquisitiones Arithmeticae

  Results from FactBites:
 
Disquisitiones Arithmeticae - Wikipedia, the free encyclopedia (573 words)
The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24.
The Disquisitiones covers both elementary number theory and parts of the area of mathematics that we now call algebraic number theory.
The Disquisitiones was one of the last mathematical works to be written in scholarly Latin (an English translation was not published until 1965).
Carl Friedrich Gouss (2791 words)
Some of his early discoveries were published in his Helmstadt dissertation of 1799 and in the impressive "Disquisitiones arithmeticae" of 1801.
The "Disquisitiones arithmeticae" collected all the masterful work in number theory of Gauss' predecessors and enriched it to such an extent that the beginning of modern number theory is sometimes dated from the publication of this book.
It was a continuation of his theory of quadratic residues in the "Disquisitiones arithmeticae," but a continuation with the aid of a new method, the theory of complex numbers.
  More results at FactBites »

 
 

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