Editing Quadratic reciprocity (section)

==Higher powers==
{{further|Cubic reciprocity|Quartic reciprocity|Octic reciprocity|Eisenstein reciprocity}}
The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including [[Carl Friedrich Gauss]], [[Peter Gustav Lejeune Dirichlet]], [[Carl Gustav Jakob Jacobi]], [[Gotthold Eisenstein]], [[Richard Dedekind]], [[Ernst Kummer]], and [[David Hilbert]] to the study of general algebraic number fields and their rings of integers;<ref>Lemmermeyer, p. 15, and Edwards, pp.79&ndash;80 both make strong cases that the study of higher reciprocity was much more important as a motivation than Fermat's Last Theorem was</ref> specifically Kummer invented ideals in order to state and prove higher reciprocity laws.

The [[Hilbert's ninth problem|ninth]] in the list of [[Hilbert's problems|23 unsolved problems]] which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the 
"Proof of the most general reciprocity law [f]or an arbitrary number field".<ref>Lemmermeyer, p. viii</ref> Building upon work by [[Philipp Furtwängler]], [[Teiji Takagi]], [[Helmut Hasse]] and others, Emil Artin discovered [[Artin reciprocity]] in 1923, a general theorem for which all known reciprocity laws are special cases, and proved it in 1927.<ref>Lemmermeyer, p. ix ff</ref>