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State/Quadratic reciprocity

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Theorem: (Quadratic Reciprocity) Suppose that $p$ and $q$ are distinct odd prime numbers. Then, the two Legendre symbols $\left( \frac{p}{q} \right)$ and $\left( \frac{q}{p} \right)$ are related by the following formula: $$\left( {p \over q} \right) \left( {q \over p} \right) = (-1)^{ { {p-1} \over 2} { {q-1} \over 2} }.$$

A convenient way to rephrase this formula is the following:

  • If $p$ and $q$ are distinct odd prime numbers, then
    • If $p \equiv 3$, mod $4$, AND $q \equiv 3$, mod $4$, then $\left( {p \over q} \right) = - \left( {q \over p} \right)$.
    • Otherwise, the two Legendre symbols are equal: $\left( {p \over q} \right) = \left( {q \over p} \right)$.

Logical Connections

This statement logically relies on the following definitions and statements: Def/Prime number, Def/Legendre symbol, State/Chinese remainder theorem, Def/Permutation, Def/Sign of a permutation, Def/Cyclic permutation, Def/Fixed point, State/Zolotarevs lemma, State/Counting subsets of a given cardinality

The following statements and definitions rely on the material of this page:

To visualize the logical connections between this statements and other items of mathematical knowledge, you can visit the following cluster(s), and click the "Visualize" tab: Clust/Theory of quadratic residues


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