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Disquisitiones Arithmeticae
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The Disquisitiones Arithmeticae 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 such as Fermat, Euler, Lagrange and Legendre. The Disquisitiones covers both elementary number theory and parts of the area of mathematics now called number theory. However, Gauss did not explicitly recognize the concept of a group and his own title for his subject was Higher Arithmetic. Gauss also states When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work. Although few of the results in these first sections are original, Gauss was the first mathematician to bring this material together and he also realized the importance of the property of unique factorization, which he restates and proves using modern tools. 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 and ternary quadratic forms. Section VI includes two different primality tests, 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 eighth section was published as a treatise entitled general investigations on congruences. Its worth notice since Gauss attacked the problem of general congruences from a standpoint closely related to that taken later by Dedekind, Galois, the treatise paved the way for the theory of function fields over a finite field of constants. Ideas unique to that treatise are clear recognition of the importance of the Frobenius morphism, the Disquisitiones was one of the last mathematical works to be written in scholarly Latin. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems, 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 set a standard for later texts, while recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples. The Disquisitiones was the point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet. Many of the annotations given by Gauss are in effect announcements of further research of his own and they must have appeared particularly cryptic to his contemporaries, they can now be read as containing the germs of the theories of L-functions and complex multiplication, in particular. Gauss Disquisitiones continued to influence in the 20th century. This was later interpreted as the determination of imaginary quadratic fields with even discriminant and class number 1,2 and 3. Sometimes referred to as the number problem, this more general question was eventually confirmed in 1986