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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