Quadratic reciprocity
Revision as of 22:09, 25 September 2008 by Boy Soprano II (talk | contribs) (cleaned the LaTeX a bit; I'll add a proof later)
Let be a prime, and let
be any integer. Then we can define the Legendre symbol
We say that is a quadratic residue modulo
if there exists an integer
so that
.
Equivalently, we can define the function as the unique nonzero multiplicative homomorphism of
into
.
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
.
- $\genfrac{(}{)}{}{}{ab}{p}\right) = \genfrac{(}{)}{}{}{a}{p} \genfrac{(}{)}{}{}{b}{p}$ (Error compiling LaTeX. Unknown error_msg).
There also exist quadratic reciprocity laws in other rings of integers. (I'll put that here later if I remember.)