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.
