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Quartic reciprocity
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Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p (mod q) to that of x4 ≡ q (mod p).
Euler made the first conjectures about biquadratic reciprocity. Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37. The first published proofs were by Eisenstein.
Since then a number of other proofs of the classical (Gaussian) version have been found, as well as alternate statements. Lemmermeyer states that there has been an explosion of interest in the rational reciprocity laws since the 1970s.[A]
A quartic or biquadratic residue (mod p) is any number congruent to the fourth power of an integer (mod p). If x4 ≡ a (mod p) does not have an integer solution, a is a quartic or biquadratic nonresidue (mod p).
As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to positive, odd primes.
The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 3 (mod 4) then a residue r is a quadratic residue (mod q) if and only if it is a biquadratic residue (mod q). Indeed, the first supplement of quadratic reciprocity states that −1 is a quadratic nonresidue (mod q), so that for any integer x, one of x and −x is a quadratic residue and the other one is a nonresidue. Thus, if r ≡ a2 (mod q) is a quadratic residue, then if a ≡ b2 is a residue, r ≡ a2 ≡ b4 (mod q) is a biquadratic residue, and if a is a nonresidue, −a is a residue, −a ≡ b2, and again, r ≡ (−a)2 ≡ b4 (mod q) is a biquadratic residue.
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 4).
Gauss proved that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p) can be divided into four sets, each containing (p−1)/4 numbers. Let e be a quadratic nonresidue. The first set is the quartic residues; the second one is e times the numbers in the first set, the third is e2 times the numbers in the first set, and the fourth one is e3 times the numbers in the first set. Another way to describe this division is to let g be a primitive root (mod p); then the first set is all the numbers whose indices with respect to this root are ≡ 0 (mod 4), the second set is all those whose indices are ≡ 1 (mod 4), etc. In the vocabulary of group theory, the first set is a subgroup of index 4 (of the multiplicative group Z/pZ×), and the other three are its cosets.
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Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p (mod q) to that of x4 ≡ q (mod p).
Euler made the first conjectures about biquadratic reciprocity. Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37. The first published proofs were by Eisenstein.
Since then a number of other proofs of the classical (Gaussian) version have been found, as well as alternate statements. Lemmermeyer states that there has been an explosion of interest in the rational reciprocity laws since the 1970s.[A]
A quartic or biquadratic residue (mod p) is any number congruent to the fourth power of an integer (mod p). If x4 ≡ a (mod p) does not have an integer solution, a is a quartic or biquadratic nonresidue (mod p).
As is often the case in number theory, it is easiest to work modulo prime numbers, so in this section all moduli p, q, etc., are assumed to positive, odd primes.
The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 3 (mod 4) then a residue r is a quadratic residue (mod q) if and only if it is a biquadratic residue (mod q). Indeed, the first supplement of quadratic reciprocity states that −1 is a quadratic nonresidue (mod q), so that for any integer x, one of x and −x is a quadratic residue and the other one is a nonresidue. Thus, if r ≡ a2 (mod q) is a quadratic residue, then if a ≡ b2 is a residue, r ≡ a2 ≡ b4 (mod q) is a biquadratic residue, and if a is a nonresidue, −a is a residue, −a ≡ b2, and again, r ≡ (−a)2 ≡ b4 (mod q) is a biquadratic residue.
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 4).
Gauss proved that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p) can be divided into four sets, each containing (p−1)/4 numbers. Let e be a quadratic nonresidue. The first set is the quartic residues; the second one is e times the numbers in the first set, the third is e2 times the numbers in the first set, and the fourth one is e3 times the numbers in the first set. Another way to describe this division is to let g be a primitive root (mod p); then the first set is all the numbers whose indices with respect to this root are ≡ 0 (mod 4), the second set is all those whose indices are ≡ 1 (mod 4), etc. In the vocabulary of group theory, the first set is a subgroup of index 4 (of the multiplicative group Z/pZ×), and the other three are its cosets.