Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he 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.

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. In his Preface to the Disquisitiones Gauss describes the scope of the book as follows:

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

Gauss also states "When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work." ("Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis brevitatis studio tribuendum est, cui quantum fieri poterat consulere oportebat")

The book is divided into seven sections, which are:

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 also realized the importance of the property of unique factorization (assured by the fundamental theorem of arithmetic, first studied by Euclid), which he restates and proves using modern tools.

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