Editing Algebraic number theory (section)

===Hilbert===
[[David Hilbert]] unified the field of algebraic number theory with his 1897 treatise ''[[Zahlbericht]]'' (literally "report on numbers"). He also resolved a significant number-theory [[Waring's problem|problem formulated by Waring]] in 1770. As with [[#The finiteness theorem|the finiteness theorem]], he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.<ref>{{citation |last=Reid |first=Constance |year=1996 |title=Hilbert |publisher=[[Springer Science and Business Media|Springer]] |isbn=0-387-94674-8}}</ref> He then had little more to publish on the subject; but the emergence of [[Hilbert modular form]]s in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on [[class field theory]]. The concepts were highly influential, and his own contribution lives on in the names of the [[Hilbert class field]] and of the [[Hilbert symbol]] of [[local class field theory]]. Results were mostly proved by 1930, after work by [[Teiji Takagi]].<ref>This work established Takagi as Japan's first mathematician of international stature.</ref>