
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (''p''/''q'') generalizing the quadratic reciprocity symbol, that describes when a prime number is an ''n''th power residue modulo another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). Hilbert reformulated the reciprocity laws as saying that a product over ''p'' of Hilbert norm residue symbols (''a'',''b''/''p''), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic Kgroups, and their relationship with the original quadratic reciprocity law can be hard to see. ==Quadratic reciprocity== (詳細はLegendre symbol, the law of quadratic reciprocity for positive odd primes states :$\backslash left(\backslash frac\backslash right)\; \backslash left(\backslash frac\backslash right)\; =\; (1)^\backslash frac\}.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Reciprocity law」の詳細全文を読む スポンサード リンク
