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).
In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas.
[4][5][6][7] Since then a number of other proofs of the classical (Gaussian) version have been found,[8] as well as alternate statements.
Lemmermeyer states that there has been an explosion of interest in the rational reciprocity laws since the 1970s.
[16] If p = a2 + b2 where a is odd and b is even, Gauss proved[17] that 2 belongs to the first (respectively second, third, or fourth) class defined above if and only if b ≡ 0 (resp.
Dirichlet[18] 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).
[24] There are a number of equivalent ways of stating Burde's rational biquadratic reciprocity law.
Then e2 = p f2 + q g2 has non-trivial integer solutions, and[30] Let p ≡ q ≡ 1 (mod 4) be primes and assume p = r2 + q s2.
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.
[34] [bold in the original]These numbers are now called the ring of Gaussian integers, denoted by Z[i].
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 imaginary root of the equation h3 = 1 ... and similarly the theory of residues of higher powers leads to the introduction of other imaginary quantities.
[35]The numbers built up from a cube root of unity are now called the ring of Eisenstein integers.
The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers of the cyclotomic number fields; the Gaussian and Eisenstein integers are the simplest examples of these.
[36] This is where the terms unit, associate, norm, and primary were introduced into mathematics.
If λ and μ are two Gaussian integers, Nλμ = Nλ Nμ; in other words, the norm is multiplicative.
The square root of the norm of λ, a nonnegative real number which may not be a Gaussian integer, is the absolute value of lambda.
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[40] an odd number to be primary if it is ≡ 1 (mod (1 + i)3).
The notions of congruence[43] and greatest common divisor[44] are defined the same way in Z[i] as they are for the ordinary integers Z.
Gauss proves the analogue of Fermat's theorem: if α is not divisible by an odd prime π, then[45] Since Nπ ≡ 1 (mod 4),
This unit is called the quartic or biquadratic residue character of α (mod π) and is denoted by[46][47] It has formal properties similar to those of the Legendre symbol.
As in that case, if the "denominator" is composite, the symbol can equal one without the congruence being solvable: Gauss stated the law of biquadratic reciprocity in this form:[2][51] Let π and θ be distinct primary primes of Z[i].
Then 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 odd relatively prime nonunits.
[52] Probably the most well-known statement is: Let π and θ be primary relatively prime nonunits.
With this normalization, the law takes the form[57] Let α = a + bi and β = c + di where a ≡ c ≡ 1 (mod 4) and b and d are even be relatively prime nonunits.
[58] Let α = a + 2bi and β = c + 2di where a and c are odd be relatively prime nonunits.
Then if λ and μ are relatively prime nonunits, Eisenstein proved[60] The references to the original papers of Euler, Dirichlet, and Eisenstein were copied from the bibliographies in Lemmermeyer and Cox, and were not used in the preparation of this article.
182–283 of The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76.
Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art.
These two papers by Franz Lemmermeyer contain proofs of Burde's law and related results: