PlanetMath: quadratic reciprocity rule

(more info)

Math for the people, by the people.

Main Menu

quadratic reciprocity rule

(Theorem)

Theorem 1 (Law of Quadratic Reciprocity) Let and be two distinct odd primes. Then:

$\displaystyle \left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{(p-1)(q-1)/4}$

where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol (or Legendre symbol).

The following is an equivalent formulation of the Law of Quadratic Reciprocity:

Theorem 2 (Quadratic Reciprocity (second form)) Let be distinct odd primes. Then:

$\displaystyle \left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$ if one of or is congruent to modulo ;
$\displaystyle \left(\frac{p}{q}\right) = - \left(\frac{q}{p}\right)$ if both and are congruent to modulo .

"quadratic reciprocity rule" is owned by alozano. [ full author list (3) | owner history (3) ]

(view preamble)

Other names:

quadratic reciprocity

Attachments:: proof of quadratic reciprocity rule (Proof) by mathcam
Zolotarev's lemma (Theorem) by mathcam
quadratic character of 2 (Theorem) by mathcam

Log in to rate this entry.
(view current ratings)

Cross-references: congruent, equivalent, Legendre symbol, primes, odd
There are 10 references to this entry.

This is version 12 of quadratic reciprocity rule, born on 2001-08-13, modified 2007-06-19.
Object id is 12, canonical name is QuadraticReciprocityRule.
Accessed 8319 times total.

Classification:

11A15 (Number theory :: Elementary number theory :: Power residues, reciprocity)

Pending Errata and Addenda

None.[ View all 8 ]

Discussion

forum policy

No messages.

Interact