# Difference between revisions of "Quadratic reciprocity"

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There are three parts. Let <math>p</math> and <math>q</math> be distinct [[odd integer | odd]] primes. Then the following hold: | There are three parts. Let <math>p</math> and <math>q</math> be distinct [[odd integer | odd]] primes. Then the following hold: | ||

− | * <math>\left(\frac{-1}{p}\right)=(-1)^{(p-1)/ | + | * <math>\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}</math>. |

* <math>\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8}</math>. | * <math>\left(\frac{2}{p}\right)=(-1)^{(p^2-1)/8}</math>. | ||

− | * <math>\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)/ | + | * <math>\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{(p-1)/2\ (q-1)/2}</math>. |

This theorem can help us evaluate Legendre symbols, since the following laws also apply: | This theorem can help us evaluate Legendre symbols, since the following laws also apply: |

## Revision as of 15:58, 3 December 2007

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.)