Higher Powers
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, J. P. G. 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; specifically Kummer invented ideals in order to state and prove higher reciprocity laws.
The ninth in the list of 23 unsolved problems which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the "Proof of the most general reciprocity law or an arbitrary number field". In 1923 Artin, building upon work by Furtwängler, Takagi, Hasse and others, discovered a general theorem for which all known reciprocity laws are special cases; he proved it in 1927.
The links below provide more detailed discussions of these theorems.
Read more about this topic: Quadratic Reciprocity
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