Difference between revisions of "Quadratic reciprocity"
m (→Quadratic Reciprocity Theorem) |
m |
||
Line 1: | Line 1: | ||
− | Let <math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer not divisible by <math>p</math>. Then we can define the [[Legendre symbol]] <math>\left(\frac{a}{p}\right)= | + | Let <math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer not divisible by <math>p</math>. Then we can define the [[Legendre symbol]] <math>\left(\frac{a}{p}\right)= |
+ | |||
+ | We say that <math>a</math> is a '''quadratic residue''' modulo <math>p</math> if there exists an integer <math>n</math> so that <math>n^2\equiv a\pmod p</math>. We can then define <math>\left(\frac{a}{p}\right)=0</math> if <math>a</math> is divisible by <math>p</math>. | ||
== Quadratic Reciprocity Theorem == | == Quadratic Reciprocity Theorem == |
Revision as of 18:46, 18 October 2006
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. 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.)