Editing Algebraic number theory (section)

===Reciprocity laws===
{{Main|Reciprocity law}}

In terms of the [[Legendre symbol]], the law of quadratic reciprocity for positive odd primes states
:<math> \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>

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 [[Legendre symbol|quadratic reciprocity symbol]], that describes when a [[prime number]] is an ''n''th power residue [[modular arithmetic|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 symbols (''a'',''b''/''p''), taking values in roots of unity, is equal to 1. [[Emil Artin|Artin]]'s reformulated [[Artin reciprocity law|reciprocity law]] states 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 K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.