Hilbert Symbol - Hilbert Symbols Over The Rationals

Hilbert Symbols Over The Rationals

For a place v of the rational number field and rational numbers a, b we let (a, b)v denote the value of the Hilbert symbol in the corresponding completion Qv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place then the completion is the real number field.

Over the reals, (a, b) is +1 if at least one of a or b is positive, and −1 if both are negative.

Over the p-adics with p odd, writing and, where u and v are integers coprime to p, we have

, where

and the expression involves two Legendre symbols.

Over the 2-adics, again writing and, where u and v are odd numbers, we have

, where .

It is known that if v ranges over all places, (a, b)v is 1 for almost all places. Therefore the following product formula

makes sense. It is equivalent to the law of quadratic reciprocity.

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