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 memoirs 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 1970's. (Here "rational" means laws that are stated in terms of ordinary integers rather in terms of the integers of some algebraic number field.)
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) every quadratic residue is also a biquadratic residue; by the first supplement of quadratic reciprocity −1 is a quadratic nonresidue (mod q), and since modulo any prime the product of two quadratic nonresidues is a residue, if x is not ≡ 0 (mod q) then one of x, −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), and we will restrict ourselves to that case.
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 third 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.
The first set is the biquadratic residues, the third set is the quadratic residues that are not quartic residues, and the second and fourth sets are the quadratic nonresidues. Gauss proved that −1 is a biqudratic residue if p ≡ 1 (mod 8) and a quadratic, but not biquadratic, residue, when p ≡ 5 (mod 8).
2 is a quadratic residue mod p if and only if p ≡ ±1 (mod 8). Since we're only considering p ≡ 1 (mod 4), this means p ≡ 1 (mod 8). Every such prime is the sum of a square and twice a square.
Let q = a2 + 2b2 ≡ 1 (mod 8) be a prime number. Then 2 is a biquadratic residue (mod q) if and only if a ≡ ±1 (mod 8); if a ≡ ±3 (mod 8) then 2 is quadratic, but not a biquadratic, residue (mod q).
2 belongs to the first (respectively second, third, or fourth) class defined above if and only if b ≡ 0 (resp. 2, 4, or 6) (mod 8). The first case of this is Euler's conjecture:
2 is a biquadratic residue of a prime p ≡ 1 (mod 4) if and only if p = a2 + 64b2.
For an odd prime number p and a quadratic residue a (mod p), Euler's criterion states that
Thus, if p ≡ 1 (mod 4),
Define the rational quartic residue symbol for prime p ≡ 1 (mod 4) and biquadratic residue a (mod p) as It is easy to prove that a is a biquadratic residue (mod p) if and only if
Dirichlet simplified Gauss's proof of the biquadratic character of 2 (his proof only requires quadratic reciprocity for the integers) and put the result in the following form:
Let p = a2 + b2 ≡ 1 (mod 4) be prime, and let i ≡ b/a (mod p). Then
- (Note that i2 ≡ −1 (mod p).)
In fact, let p = a2 + b2 = c2 + 2d2 = e2 − 2f2 ≡ 1 (mod 8) be prime, and assume a is odd. Then
- where is the ordinary Legendre symbol.
Going beyond the character of 2, let the prime p = a2 + b2 where b is even, and let q be a prime such that Quadratic reciprocity says that where Let σ2 ≡ p (mod q). Then
- This implies that
The first few examples are:
Dirichlet also proved that if p ≡ 1 (mod 4) is prime and then
This has been extended from 17 to 17, 73, 97, and 193 by Brown and Lehmer.
There are a number of equivalent ways of stating Burde's rational biquadratic reciprocity law.
They all assume that p = a2 + a2 and q = c2 + d2 are primes where b and d are even, and that
Gosset's version is
Letting i2 ≡ −1 (mod p) and j2 ≡ −1 (mod q), Frölich's law is
Burde stated his in the form:
Let p ≡ q ≡ 1 (mod 4) be primes and assume . Then e2 = p f2 + q g2 has non-trivial integer solutions, and
Let p ≡ q ≡ 1 (mod 4) be primes and assume p = r2 + q s2. Then
Let p = 1 + 4x2 be prime, let a be any odd number that divides x, and let Then a* is a biquadratic residue (mod p).
Let p = a2 + 4b2 = c2 + 2d2 ≡ 1 (mod 8) be prime. Then all the divisors of c4 − p a2 are biquadratic residues (mod p). The same is true for all the divisors of d4 − p b2.
In his second monograph on biquadratic reciprocity Gauss displays some examples and makes conjectures that imply the theorems listed above for the biquadratic character of small primes. He makes some general remarks, and admits there is no obvious general rule at work. He goes on to say
The theorems on biquadratic residues gleam with the greatest simplilcity and genuine beauty only when the field of arithmetic is extended to imaginary numbers, so that without restriction, the numbers of the form a + bi constitute the object of study ... we call such numbers integral complex numbers.
In a footnote he adds
The theory of cubic residues must be based in a similar way on a consideration of numbers of the form a' + bh where h is an imaaginary root of the equation h3 = 1 ... and similarly the theory of residues of higher powers leads to thte introduction of other imaginary quamntities.
The numbers built up from a cube root of unity are now called the ring of Eisenstein integers.
Gauss develops the arithmetic theory of the "integral complex numbers" and shows that it is quite similar to the arithmetic of ordinary integers. This is where the terms unit, associate, norm, and primary were introduced into mathematics.
The units are the numbers that divide 1. They are 1, i, −1, and −i. They are similar to 1 and −1 in the ordinary integers, in that they divide evey number. The units are the powers of i.
Given a number λ = a + bi, its conjugate is a − bi and its associates are the four numbers
λ = +a + bi
iλ = −b + ai
−λ = −a − bi
−iλ = +b − ai
The norm of λ = a + bi is the number Nλ = a2 + b2. If λ and μ are two Gaussian integers, Nλμ = Nλ Nμ; in other words, the norm is multiplicative. The norm of zero is zero, the norm of any other number is a positive integer.
- 2 is a special case: 2 = i3 (i + i)2. In algebraic number theory, 2 is said to ramify in Z[i].
- Positive primes in Z ≡ 3 (mod 4) are also primes in Z[i]. In algebraic number theory, these primes are said to remain inert in Z[i].
- Positive primes in Z ≡ 1 (mod 4) are the product of two conjugate primes in Z[i]. In algebraic number theory, these primes are said to split in Z[i].
Thus, inert primes are 3, 7, 11, 19, ... and the fatorization of the split primes is 5 = (2 + i)(2 − i), 13 = (2 + 3i)(2 − 3i), 17 = (4 + i)(4 − i), 29 = (2 + 5i)(2 − 5i), ...
The associates of a prime are also primes.
Note that the norm of an inert prime q is Nq = q2 ≡ 1 (mod 4); thus the norm of all primes other than 1 + i and its associates is ≡ 1 (mod 4).
Gauss calls a number in Z[i] odd if its norm is an odd integer. Thus all primes except 1 + i and its associates are odd.
In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Gauss defines an odd number to be primary if it is ≡ 1 (mod (1 + i)3). It is straightforward to show that every odd number has exactly one primary associate. An odd number λ = a + bi is primary if a + b ≡ a − b ≡ 1 (mod 4); i.e., a ≡ 1 and b ≡ 0, or a ≡ 3 and b ≡ 2 (mod 4).
The unique factorization theorem for Z[i] is: if λ ≠ 0, then
where 0 ≤ μ ≤ 3, ν ≥ 0, the πis are primary primes and the αis ≥ 1, and this representation is unique, up to the order of the factors.
Since Nπ ≡ 1 (mod 4), is well-defined, and it is easy to show that for a unique unit ik.
This unit is called the quartic or biquadratic residue character (mod π) and is denoted by
- Its usefulness comes from the theorem
The congruence is solvable in Z[i] if and only if
The biquadratic character can be extended to odd, but not necessarily prime, numbers in the "denominator" in the same way the Legendre symbol is generalized into the Jacobi symbol. As in that case, if the "denominator" is composite, the symbol can equal one without the conguence being solvable:
Let π and θ be distinct primary primes of Z[i]. Then
- if either π or θ or both are ≡ 1 (mod 4), then but
- if both π and θ are ≡ 3 + 2i (mod 4), then
Another way to state it is
if π = a + bi is a primary prime, then
In fact, just as the quadratic reciprocity law for the Legendre symbol is also true for the Jacobi symbol, the requirement that the numbers be prime is not needed; it suffices that they be relatively prime.
Jacobi defined π = a + bi to be primary if a ≡ 1 (mod 4). With this normalization, the law takes the form
Let α = a + bi and β = c + di where a and c are ≡ 1 (mod 4) and b and d are even be relatively prime. Then
The following version was found in Gauss's unpublished manuscripts.
Let α = a + 2bi and β = c + 2di where a and c are odd be relatively prime. Then
There is a form of the law that does not use any concept of primary:
If λ is odd, let ε(λ) be the unique unit congruent to λ (mod (1 + 2i)3); i.e., ε(λ) = ik ≡ λ (mod 2 + 2i), where 0 ≤ k ≤ 3. Then for odd and relatively prime α and β,
- Euler, Tractatus, § 456
- Gauss, BQ, § 67
- Lemmermeyer, p. 200
- Eisenstein, Lois de reciprocite
- Eisenstein, Einfacher Beweis ...
- Eisenstein, Application de l'algebre ...
- Eisenstein, Beitrage zur Theorie der elliptischen ...
- Lemmermeyer, pp. 199–202
- Lemmermeyer, p. 172
- Gauss, BQ § 2
- Gauss, BQ § 2
- Gauss, BQ § 3
- Gauss, BQ §§ 4–7
- Gauss, BQ § 8
- Gauss, BQ § 10
- Gauss, DA Art. 182
- Gauss, BQ § 10
- Gauss, DA, Art. 182
- Gauss BQ §§ 14–21
- Dirichlet, Demonstration ...
- Lemmermeyer, Prop. 5.4
- Lemmermeyer, Prop. 5.5
- Lemmermeyer, Ex. 5.6
- Lemmmermeyer, pp.159, 190
- Dirichlet, Untersuchungen ...
- Lemmermeyer, Ex. 5.19
- Lemmermeyer, p. 172
- Lemmermeyer, p. 173
- Lemmermeyer, p. 167; Ireland & Rosen pp.128–130
- Lemmermeyer, Ex. 5.13
- Lemmermeyer, Ex. 5.5
- Lemmermeyer, Ex. 5.6, credited to Brown
- Lemmermeyer, Ex. 6.5, credited to Sharifi
- Lemmermeyer, Ex. 6.11, credited to E. Lehmer
- Gauss, BQ, § 30, translation in Cox, p. 83
- Gauss, BQ, § 30, translation in Cox, p. 84
- Gauss, BQ, §§ 30–55
- Gauss, BQ, § 31
- Gauss, BQ, § 31
- Gauss, BQ, § 31
- Gauss, BQ, §§ 33–34
- Gauss, BQ, § 35. He defines "halfeven" numbers as those divisible by 1 + i but not by 2, and "even" numbers as those divisible by 2.
- Gauss, BQ, § 36
- Gauss, BQ, § 37
- Gauss, BQ, §§ 38–45
- Gauss, BQ, §§ 46–47
- Gauss, BQ, § 51
- Gauss defined the character as the exponent k rather than the unit ik; also, he had no symbol for the character.
- Gauss, BQ, § 61
- Gauss, BQ, § 67
- proofs are in Lemmermeyer, chs. 6 and 8, Ireland & Rosen, ch. 9.7–9.10
- Lemmermeyer, ch. 6, Ireland & Rosen ch. 9.7–9.10
- Lemmermeyer, Th. 6.9; Ireland & Rosen, Ex. 9.32–9.37
- Gauss proves the law for 1 + i in BQ, §§ 68–76
- Lemmermeyer, Th. 69.
- Lemmermeyer, Th. 6.9
- Lemmermeyer, Ex. 6.17
- Lemmermeyer, Ex. 6.18 and p. 275
- Euler, Leonhard (1849), Tractatus de numeroroum doctrina capita sedecim quae supersunt, Comment. Arithmet. 2
This was actually written 1748–1750, but was only published posthumously; It is in Vol V, pp. 182–283 of
- Euler, Leonhard (1911–1944), Opera Omnia, Series prima, Vols I–V, Leipzig & Berlin: Teubner Check date values in:
The two monographs Gauss published on biquadratic reciprocity have consecutively-numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n".
- Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6
- Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7
These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148
German translations are in pp. 511–533 and 534–586 of the following, which also has the Disquisitiones Arithmeticae and Gauss's other papers on number theory.
- Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithmeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8
- Eisenstein, Ferdinand Gotthold (1844), Lois de réciprocité, J. Reine Angew. Math. 28, pp. 53–67 (Crelle's Journal)
- Eisenstein, Ferdinand Gotthold (1844), Einfacher Beweis und Verallgemeinerung des Fundamentaltheorems für die biquadratischen Reste, J. Reine Angew. Math. 28 pp. 223–245 (Crelle's Journal)
- Eisenstein, Ferdinand Gotthold (1845), Application de l'algèbre à l'arithmétique transcendante, J. Reine Angew. Math. 29 pp. 177–184 (Crelle's Journal)
- Eisenstein, Ferdinand Gotthold (1846), Beiträge zur Theorie der elliptischen Funktionen I: Ableitung des biquadratischen Fundalmentaltheorems aus der Theorie der Lemniskatenfunctionen, nebst Bemerkungen zu den Multiplications- und Transformationsformeln, J. Reine Angew. Math. 30 pp. 185–210 (Crelle's Journal)
These papers are all in Vol I of his Werke.
- Dirichlet, Pierre Gustave LeJeune (1832), Démonstration d'une propriété analogue à la loi de Réciprocité qui existe entre deux nombres premiers quelconques, J. Reine Angew. Math. 9 pp. 379–389 (Crelle's Journal)
- Dirichlet, Pierre Gustave LeJeune (1833), Untersuchungen über die Theorie der quadratischen Formen, Abh. Königl. Preuss. Akad. Wiss. pp. 101–121
both of these are in Vol I of his Werke.
- Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (Second edition), New York: Springer, ISBN 0-387-97329-X
- Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66967-4 Check
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These two papers by Franz Lemmermeyer contain proofs of Burde's law and related results;