Difference between revisions of "Quadratic residues"
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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]]. | ||
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Revision as of 22:45, 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.
