Quadratic reciprocity
Revision as of 17:15, 12 July 2006 by ComplexZeta (talk | contribs)
Let be a prime, and let be any integer not divisible by . Then we can define the Legendre symbol We say that is a quadratic residue modulo if there exists an integer so that . We can then define if is divisible by .
Quadratic Reciprocity Theorem
There are three parts. Let and be distinct odd primes. The the following hold:
- .
- .
- .
This theorem can help us evaluate Legendre symbols, since the following laws also apply:
- If , then .
- .
There also exist quadratic reciprocity laws in other rings of integers. (I'll put that here later if I remember.)