Difference between revisions of "Quadratic Reciprocity Theorem"

Latest revision as of 22:21, 5 April 2021

Quadratic reciprocity is a classic result of number theory.
It is one of the most important theorems in the study of quadratic residues.

Statement

It states that $\left(\frac{p}{q}\right)= \left(\frac{q}{p}\right)$ for primes $p$ and $q$ greater than $2$ where both are not of the form $4n+3$ for some integer $n$ .
If both $p$ and $q$ are of the form $4n+3$ , then $\left(\frac{p}{q}\right)= -\left(\frac{q}{p}\right).$

Another way to state this is:
$\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}.$

Note that $\left(\frac{p}{q}\right)$ is not a fraction. It is the Legendre notation of quadratic residuary.

@@ Line 14: / Line 14: @@
 [[Category:Number theory]]
 [[Category:Theorems]]
-[[theorem|Quadratic Residues]]
+[[Quadratic Residues |Quadratic Residues]]

Page

Toolbox

Search

Difference between revisions of "Quadratic Reciprocity Theorem"

Latest revision as of 22:21, 5 April 2021

Statement

See Also