Hilbert Symbol

For any two nonzero p-adic numbers  and , the Hilbert symbol is defined as

(1)

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.