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> 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> | + | 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 [b]not[/b] a fraction. It is the [b]Legendre notation[/b] of quadratic residuary. | ||
==See Also== | ==See Also== | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
Revision as of 22:18, 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 [b]not[/b] a fraction. It is the [b]Legendre notation[/b] of quadratic residuary.
is [b]not[/b] a fraction. It is the [b]Legendre notation[/b] of quadratic residuary.
