Quadratic reciprocity
Revision as of 17:10, 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 $\left(\frac{a}{p}\right)=
Quadratic Reciprocity Theorem
There are three parts. Let and be distinct odd primes. The the following hold:
- .
- $\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8)$ (Error compiling LaTeX. Unknown error_msg).
- .
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.)