Quadratic reciprocity
Revision as of 17:15, 12 July 2006 by ComplexZeta (talk | contribs) (→Quadratic Reciprocity Theorem)
Let be a prime, and let
be any integer not divisible by
. Then we can define the Legendre symbol $\left(\frac{a}{p}\right)=
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.)