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In mathematics, the law of quadratic reciprocity in number theory, 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
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has a solution x if and only if the congruence
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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
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has a solution x if and only if the congruence
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does not have a solution y.
Using the Legendre symbol (p/q), these
statements may be summarized as
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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
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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 Lemma of Gauss reasons about the properties of quadratic residues and is involved in Gauss's proof of quadratic
reciprocity.
External links
- A play comparing two proofs of the quadratic reciprocity
law
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