The Disquisitiones Arithmeticae Latin for "Arithmetical Investigations" is a textbook of number theory written in Latin [1] by Carl Friedrich Gauss in when Gauss was 21 and first published in when he was It is notable for having had a revolutionary impact on the field of number theory as it not only made the field truly rigorous and systematic but also paved the path for modern number theory. In this book Gauss brought together and reconciled results in number theory obtained by mathematicians such as Fermat , Euler , Lagrange , and Legendre and added many profound and original results of his own. The Disquisitiones covers both elementary number theory and parts of the area of mathematics now called algebraic number theory. However, Gauss did not explicitly recognize the concept of a group , which is central to modern algebra , so he did not use this term. His own title for his subject was Higher Arithmetic.

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Leipzig: Gerh. Fleischer, In this book Gauss standardized the notation; he systemized the existing theory and extended it; and he classified the problems to be studied and the known methods of attack and introduced new methods … The Disquisitiones not only began the modern theory of numbers but determined the direction of work in the subject up to the present time. In his Disquisitiones Gauss summarized previous work in a systematic way, solved some of the most difficult outstanding questions, and formulated concepts and questions that set the pattern of research for a century and still have significant today.

He proved the law of quadratic reciprocity, developed the theory of composition of quadratic forms, and completely analyzed the cyclotomic equation. This is why a treatise as profound and as novel as his Arithmetical Investigations heralds M.

Gauss as one of the best mathematical minds in Europe. During the XIXth century, its fame grew to almost mythical dimensions. An early version of the treatise was completed a year later. In November , Gauss started rewriting the early version into the more mature text which he would give to the printer bit by bit.

Printing started in April , but proceeded very slowly for technical reasons on the part of the printer. Gauss resented this very much, as his letters show; he was looking for a permanent position from But he did use the delays to add new text, in particular to sec.

The first and smallest one 7 pp. Each of the numbers b , c are called a residue of the other in the first case, a nonresidue in the second. The remainder of sec. At the end of sec.

After a discussion, in arts. A crucial nontrivial ingredient used in art. With its pp. Leonhard Euler, Joseph-Louis Lagrange, and Adrien-Marie Legendre had forged tools to study the representation of integers by quadratic forms. Gauss, however, moved away from this Diophantine aspect towards a treatment of quadratic forms as objects in their own right, and, as he had done for congruences, explicitly pinpointed and named the key tools. This move is evident already in the opening of sec.

The first part of sec. Gauss defined two quadratic forms art. The number of reduced forms — and thus also the number of equivalence classes of forms — of a given determinant is finite … Gauss settled the general problem of representing integers by quadratic forms arts.

There follows arts. For this last application, as well as for deeper insight into the number of genera, Gauss quickly generalized art. Berlin: Haude und Spener, ] at age 15, and launched into counting prime numbers in given intervals in order to guess their asymptotic distribution. This is because there is no way of knowing beforehand whether a given number has any divisors or not. Section 6 52 pp. In the earlier part of sec. Its latter part arts. After a few reminders on circular functions … Gauss focused on the prime case and the irreducible equation.

At several places in the Disquisitiones arithmeticae and in his correspondence a forthcoming volume II is referred to. This differs from the structure of the published Disquisitiones arithmeticae in that it contains an incomplete 8 th chapter caput octavum , devoted to higher congruences, i. The inventory was last updated: 4th June An entirely unrestored copy. Add to Cart Ask a Question.

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