Difference between revisions of "Quadratic residues"
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== Legendre Symbol == | == Legendre Symbol == | ||
− | Determining whether <math>a</math> is a quadratic residue modulo <math>m</math> is easiest if <math>m=p</math> is a [[prime number|prime]]. In this case we write <math>\left(\frac{a}{p}\right)= | + | Determining whether <math>a</math> is a quadratic residue modulo <math>m</math> is easiest if <math>m=p</math> is a [[prime number|prime]]. In this case we write <math>\left(\frac{a}{p}\right)=\begin{cases} 0 & \mathrm{if }\ p\mid a, \ 1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ residue\ modulo\ }\ p, \ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math> |
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+ | The symbol <math>\left(\frac{a}{p}\right)</math> is called the [[Legendre symbol]]. | ||
== Quadratic Reciprocity == | == Quadratic Reciprocity == |
Revision as of 12:30, 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.
(I'm sure someone wants to write out all the fun properties of Legendre symbols. It just happens not to be me right now.)