In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field.
There are also several explicit reciprocity laws for various generalizations of the Hilbert symbol to higher local fields, p-divisible groups, and so on.
Artin & Hasse (1928) gave an explicit formula for the Hilbert symbol (α,β) in the case of odd prime powers, for some special values of α and β when the field is the (cyclotomic) extension of the p-adic numbers by a pnth root of unity.
Shafarevich (1950) gave an explicit formula for the Hilbert symbol for odd prime powers for general local fields.
Henniart (1981) simplified Vostokov's work and extended it to the case of even prime powers.
For archimedean local fields or in the unramified case the Hilbert symbol is easy to write down explicitly.
In the unramified case, when the order of the Hilbert symbol is coprime to the residue characteristic of the local field, the tame Hilbert symbol is given by[1] where ω(a) is the (q – 1)-th root of unity congruent to a and ord(a) is the value of the valuation of the local field, and n is the degree of the Hilbert symbol, and q is the order of the residue class field.
The number n divides q – 1 because the local field contains the nth roots of unity by assumption.