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. Then 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.)