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quadratic reciprocity rule
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(Theorem)
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Theorem 1 (Law of Quadratic Reciprocity) Let $p$ and $q$ be two distinct odd primes. Then:
$$ \left(\frac{q}{p}\right)\left(\frac{p}{q}\right)=(-1)^{(p-1)(q-1)/4} $$
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 $p,q$ be distinct odd primes. Then:
- $\displaystyle \left(\frac{p}{q}\right) = \left(\frac{q}{p}\right)$ if one of $p$ or $q$ is congruent to $1$ modulo $4$ ;
- $\displaystyle \left(\frac{p}{q}\right) = - \left(\frac{q}{p}\right)$ if both $p$ and $q$ are congruent to $3$ modulo $4$ .
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"quadratic reciprocity rule" is owned by alozano. [ full author list (3) | owner history (3) ]
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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 10335 times total.
Classification:
| AMS MSC: | 11A15 (Number theory :: Elementary number theory :: Power residues, reciprocity) |
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Pending Errata and Addenda
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