Editing Number theory (section)

=== Maturity and division into subfields ===
[[File:Peter Gustav Lejeune Dirichlet.jpg|upright=0.8|thumb|[[Peter Gustav Lejeune Dirichlet]]]]

Starting early in the nineteenth century, the following developments gradually took place:
* The rise to self-consciousness of number theory (or ''higher arithmetic'') as a field of study.<ref>See the discussion in section 5 of {{harvnb|Goldstein|Schappacher|2007}}. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted in {{harvnb|Weil|1984|p=25}}).</ref>
* The development of much of modern mathematics necessary for basic modern number theory: [[complex analysis]], [[group theory]], [[Galois theory]]—accompanied by greater rigor in analysis and abstraction in algebra.
* The rough subdivision of number theory into its modern subfields—in particular, [[analytic number theory|analytic]] and algebraic number theory.

Algebraic number theory may be said to start with the study of reciprocity and [[cyclotomy]], but truly came into its own with the development of [[abstract algebra]] and early ideal theory and [[valuation (algebra)|valuation]] theory; see below. A conventional starting point for analytic number theory is [[Dirichlet's theorem on arithmetic progressions]] (1837),{{sfn|Apostol|1976|p=7}}{{sfn|Davenport|Montgomery|2000|p=1}} whose proof introduced [[L-functions]] and involved some asymptotic analysis and a limiting process on a real variable.<ref>See the proof in {{harvnb|Davenport|Montgomery|2000|loc=section 1}}</ref> The first use of analytic ideas in number theory actually goes back to Euler (1730s),{{sfn|Iwaniec|Kowalski|2004|p=1}}{{sfn|Varadarajan|2006|loc=sections 2.5, 3.1 and 6.1}} who used formal power series and non-rigorous (or implicit) limiting arguments. The use of ''complex'' analysis in number theory comes later: the work of [[Bernhard Riemann]] (1859) on the [[Riemann zeta function|zeta function]] is the canonical starting point;{{sfn|Granville|2008|pp=322–348}} [[Jacobi's four-square theorem]] (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory ([[modular form]]s).<ref>See the comment on the importance of modularity in {{harvnb|Iwaniec|Kowalski|2004|p=1}}</ref>

The [[American Mathematical Society]] awards the ''[[Cole Prize]] in Number Theory''. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the ''[[Fermat Prize]]''.