Disquisitiones Arithmeticae - Importance

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.

The Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, J. P. G. Lejeune Dirichlet and Richard 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of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished.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.

Gauss' Disquisitiones continued to exert influence in the 20th century. For example, in section V, article 303, Gauss summarized his calculations of class numbers of proper primitive binary quadratic forms, and conjectured that he had found all of them with class numbers 1, 2, and 3. This was later interpreted as the determination of imaginary quadratic number fields with even discriminant and class number 1,2 and 3, and extended to the case of odd discriminant. Sometimes referred to as the Class number problem, this more general question was eventually confirmed in 1986 (the specific question Gauss asked was confirmed by Landau in 1902, for class number one). In section VII, article 358, Gauss proved what can be interpreted as the first non-trivial case of the Riemann hypothesis for curves over finite fields (the Hasse–Weil theorem).