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