Quadratic reciprocity

English

Alternative forms

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Etymology

The theorem highlights a particular form of reciprocity in the solvability of the quadratic equation a² = b in modular arithmetic. It was conjectured by Template:W and Template:W and first proved by Template:W.

Noun

1. Template:Lb The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p.
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Usage notes

There are several equivalent statements of the theorem. One version states that if p and q are odd prime numbers, $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1{)}^{\frac{p-1}{2}\frac{q-1}{2}}$, where $\left(\frac{p}{q}\right)$ is the Legendre symbol. This equation remains valid if $\left(\frac{p}{q}\right)$ is interpreted as a Jacobi symbol, in which case p and q are (only) required to be odd positive coprime integers. However, the value of the Jacobi symbol is less informative about whether p is a square modulo q (it can reveal that it is not, but not definitively that it it is).

The equation can be used to simplify calculation of the Legendre / Jacobi symbol.

See also

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Further reading

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