# Difference between revisions of "Quadratic reciprocity"

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− | 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)=\begin{cases} 1 & a\mathrm{\ is\ a\ quadratic\ residue\ modulo\ } p, \\ -1 & \mathrm{otherwise}.\end{cases}</math> 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>. | + | 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)=\begin{cases} 1 & a\mathrm{\ is\ a\ quadratic\ residue\ modulo\ } p, \\ -1 & \mathrm{otherwise}.\end{cases}</math> |

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+ | 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 19:46, 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.)