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Home > Quadratic reciprocity

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

has a solution x if and only if the congruence

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

has a solution x if and only if the congruence

does not have a solution y.

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

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

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).

The Gauss lemma reasons about the properties of quadratic residues and is involved in two of Gauss's proofs of quadratic reciprocity.

External link

• A play comparing two proofs of the quadratic reciprocity law

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Topics: Quadratic Reciprocity Congruence Laws Solution Arithmetic...

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