Let be a nonzero rational number , where , ..., are distinct primes, and . Then(1)(2)
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.