For any two nonzero p-adic numbers and , the Hilbert symbol is defined as
(1)
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If the -adic field is not clear, it is said to be the Hilbert symbol of and relative to . The field can also be the reals (). The Hilbert symbol satisfies the following formulas:
1. .
2. for any .
3. .
4. .
5. .
6. .
The Hilbert symbol depends only the values of and modulo squares. So the symbol is a map .
Hilbert showed that for any two nonzero rational numbers and ,
1. for almost every prime .
2. where ranges over every prime, including corresponding to the reals.