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Quadratic reciprocity

The law of quadratic reciprocity, conjectured by Euler and Legendre and first satisfactorily proved by Gauss, connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows to determine the solvability of any quadratic equation in modular arithmetic.

Suppose p and q are two different odd primes. If at least one of them is congruent to 1 modulo 4, then the congruence

<math>x^2\equiv p\ ({\rm mod}\ q)</math>
has a solution x if and only if the congruence
<math>y^2\equiv q\ ({\rm mod}\ p)</math>
has a solution y. (The two solutions will in general be different.) On the other hand, if both primes are congruent to 3 modulo 4, then the congruence
<math>x^2\equiv p\ ({\rm mod}\ q)</math>
has a solution x if and only if the congruence
<math>y^2\equiv q\ ({\rm mod}\ p)</math>
does not have a solution y.

Using the Legendre symbol (p/q), these statements may be summarized as

<math>(p/q)\cdot(q/p)=(-1)^{(p-1)/2\cdot(q-1)/2}.</math>

For example taking p to be 11 and q to be 19, we can relate (11/19) to (19/11) which is (8/11). To proceed further we may need to know the supplementary laws computing (2/q) and (-1/q) explicitly. For example

<math>(-1/q)= (-1)^{(q-1)/2}.</math>

Using this we relate (8/11) to (-3/11) to (3/11) to (11/3) to (2/3) to (-1/3); and can complete the initial calculation.

In a book about reciprocity laws published in 2000, Lemmermeyer collects literature citations for 196 different published proofs for the quadratic reciprocity law.

There are cubic, quartic (biquadratic) and other higher reciprocity laws[?]; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws).

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