Difference between revisions of "Quadratic Reciprocity Theorem"
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It is one of the most important theorems in the study of quadratic residues. | It is one of the most important theorems in the study of quadratic residues. | ||
− | 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>. | + | ==Statement== |
− | 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> | + | 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> | ||
Another way to state this is:<br> | Another way to state this is:<br> | ||
− | <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== | ||
+ | [[Category:Number theory]] | ||
+ | [[Category:Theorems]] | ||
+ | [[Quadratic Residues |Quadratic Residues]] |
Latest revision as of 23:21, 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 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.