# Difference between revisions of "Quadratic residues"

(→Legendre Symbol: cleaned up the TeX a bit: \ is your friend) |
m (Moving comment on symbols to talk page) |
||

Line 14: | Line 14: | ||

Now suppose that <math>m</math>, as above, is not [[composite]], and let <math>m=p_1^{e_1}\cdots p_n^{e_n}</math>. Then we write <math>\left(\frac{a}{m}\right)=\left(\frac{a}{p_1}\right)^{e_1}\cdots\left(\frac{a}{p_n}\right)^{e_n}</math>. This symbol is called the [[Jacobi symbol]]. | Now suppose that <math>m</math>, as above, is not [[composite]], and let <math>m=p_1^{e_1}\cdots p_n^{e_n}</math>. Then we write <math>\left(\frac{a}{m}\right)=\left(\frac{a}{p_1}\right)^{e_1}\cdots\left(\frac{a}{p_n}\right)^{e_n}</math>. This symbol is called the [[Jacobi symbol]]. | ||

− | |||

− |

## Revision as of 23:45, 24 June 2006

Let and be integers, with . We say that is a **quadratic residue** modulo if there is some number so that is divisible by .

## Legendre Symbol

Determining whether is a quadratic residue modulo is easiest if is a prime. In this case we write

The symbol is called the Legendre symbol.

## Quadratic Reciprocity

Let and be distinct odd primes. Then . This is known as the Quadratic Reciprocity Theorem.

## Jacobi Symbol

Now suppose that , as above, is not composite, and let . Then we write . This symbol is called the Jacobi symbol.