Difference between revisions of "Quadratic Reciprocity Theorem"
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==Statement== | ==Statement== | ||
| − | It states that <math>\left(\frac{p}{q}\right)= \left(\frac{q}{p}\right)</math> | + | It states that <math>\left(\frac{p}{q}\right)= \left(\frac{q}{p}\right)</math> for primes <math>p</math> and <math>q</math> greater than <math>2</math> where both are not of the form <math>4n+3</math> for some integer <math>n</math>.<br> |
If both <math>p</math> and <math>q</math> are of the form <math>4n+3</math>, then <math>\left(\frac{p}{q}\right)= -\left(\frac{q}{p}\right).</math> | If both <math>p</math> and <math>q</math> are of the form <math>4n+3</math>, then <math>\left(\frac{p}{q}\right)= -\left(\frac{q}{p}\right).</math> | ||
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<math>\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math> | <math>\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math> | ||
| − | + | Note that <math>\left(\frac{p}{q}\right)</math> is not a fraction. It is the Legendre notation of quadratic residuary. | |
==See Also== | ==See Also== | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
Revision as of 22:19, 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 
for primes 
and 
greater than 
where both are not of the form 
for some integer 
.
If both and are of the form , then 
Another way to state this is:
Note that 
is not a fraction. It is the Legendre notation of quadratic residuary.
