Background 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 simplicity 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. [bold in the original] These numbers are now called the
ring of
Gaussian integers, denoted by
Z[
i]. Note that
i is a fourth root of 1. 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. 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.
Facts and terminology 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 every 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 • 2 is a special case: 2 =
i3 (1 +
i)2. It is the only prime in
Z divisible by the square of a prime in
Z[
i]. 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 a factorization of the split primes is : 5 = (2 +
i) × (2 −
i), :13 = (2 + 3
i) × (2 − 3
i), :17 = (4 +
i) × (4 −
i), :29 = (2 + 5
i) × (2 − 5
i), ... The associates and conjugate of a prime are also primes. Note that the norm of an inert prime
q is N
q =
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. The product of two odd numbers is odd and the conjugate and associates of an odd number 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 product of two primary numbers is primary and the conjugate of a primary number is also primary. The unique factorization theorem for
Z[
i] is: if λ ≠ 0, then :\lambda = i^\mu(1+i)^\nu\pi_1^{\alpha_1}\pi_2^{\alpha_2}\pi_3^{\alpha_3} \dots where 0 ≤ μ ≤ 3, ν ≥ 0, the π
is are primary primes and the α
is ≥ 1, and this representation is unique, up to the order of the factors. The notions of
congruence and
greatest common divisor are defined the same way in
Z[
i] as they are for the ordinary integers
Z. Because the units divide all numbers, a congruence (mod λ) is also true modulo any associate of λ, and any associate of a GCD is also a GCD.
Quartic residue character Gauss proves the analogue of
Fermat's theorem: if α is not divisible by an odd prime π, then :\alpha^{N \pi - 1} \equiv 1 \pmod{\pi} Since Nπ ≡ 1 (mod 4), \alpha^{\frac{N\pi - 1}{4}} makes sense, and \alpha^{\frac{N\pi - 1}{4}}\equiv i^k \pmod{\pi} for a unique unit
ik. This unit is called the
quartic or
biquadratic residue character of α (mod π) and is denoted by :\left[\frac{\alpha}{\pi}\right] = i^k \equiv \alpha^{\frac{N\pi - 1}{4}} \pmod{\pi}. It has formal properties similar to those of the
Legendre symbol. :The congruence x^4 \equiv \alpha \pmod{\pi} is solvable in
Z[
i] if and only if \left[\frac{\alpha}{\pi}\right] = 1. :\Bigg[\frac{\alpha\beta}{\pi}\Bigg]=\Bigg[\frac{\alpha}{\pi}\Bigg]\Bigg[\frac{\beta}{\pi}\Bigg] :\overline{\Bigg[\frac{\alpha}{\pi}\Bigg]}=\Bigg[\frac{\overline{\alpha}}{\overline{\pi}}\Bigg] where the bar denotes
complex conjugation. :if π and θ are associates, \Bigg[\frac{\alpha}{\pi}\Bigg]=\Bigg[\frac{\alpha}{\theta}\Bigg] :if α ≡ β (mod π), \Bigg[\frac{\alpha}{\pi}\Bigg]=\Bigg[\frac{\beta}{\pi}\Bigg] The biquadratic character can be extended to odd composite 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 congruence being solvable: :\left[\frac{\alpha}{\lambda}\right] = \left[\frac{\alpha}{\pi_1}\right]^{\alpha_1} \left[\frac{\alpha}{\pi_2}\right]^{\alpha_2} \dots where \lambda = \pi_1^{\alpha_1}\pi_2^{\alpha_2}\pi_3^{\alpha_3} \dots :If
a and
b are ordinary integers,
a ≠ 0, |
b| > 1, gcd(
a,
b) = 1, then \left[\frac{a}{b}\right] = 1.
Statements of the theorem Gauss stated the law of biquadratic reciprocity in this form: Let π and θ be distinct primary primes of
Z[
i]. Then :if either π or θ or both are ≡ 1 (mod 4), then \Bigg[\frac{\pi}{\theta}\Bigg] =\left[\frac{\theta}{\pi}\right], but :if both π and θ are ≡ 3 + 2
i (mod 4), then \Bigg[\frac{\pi}{\theta}\Bigg] =-\left[\frac{\theta}{\pi}\right]. 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. Probably the most well-known statement is: Let π and θ be primary relatively prime nonunits. Then :\Bigg[\frac{\pi}{\theta}\Bigg]\left[\frac{\theta}{\pi}\right]^{-1}= (-1)^{\frac{N\pi - 1}{4}\frac{N\theta-1}{4}}. There are supplementary theorems for the units and the half-even prime 1 +
i. if π =
a +
bi is a primary prime, then :\Bigg[\frac{i}{\pi}\Bigg]=i^{-\frac{a-1}{2}},\;\;\; \Bigg[\frac{1+i}{\pi}\Bigg]=i^\frac{a-b-1-b^2}{4}, and thus :\Bigg[\frac{-1}{\pi}\Bigg]=(-1)^{\frac{a-1}{2}},\;\;\; \Bigg[\frac{2}{\pi}\Bigg]=i^{-\frac{b}{2}}. Also, if π =
a +
bi is a primary prime, and
b ≠ 0 then :\Bigg[\frac{\overline{\pi}}{\pi}\Bigg]=\Bigg[\frac{-2}{\pi}\Bigg](-1)^\frac{a^2-1}{8} (if
b = 0 the symbol is 0). 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 ≡
c ≡ 1 (mod 4) and
b and
d are even be relatively prime nonunits. Then :\left[\frac{\alpha}{\beta}\right]\left[\frac{\beta}{\alpha}\right]^{-1}= (-1)^{\frac{bd}{4}} The following version was found in Gauss's unpublished manuscripts. Let α =
a + 2
bi and β =
c + 2
di where
a and
c are odd be relatively prime nonunits. Then :\left[\frac{\alpha}{\beta}\right]\left[\frac{\beta}{\alpha}\right]^{-1}= (-1)^{bd+\frac{a-1}{2}d+\frac{c-1}{2}b},\;\;\;\; \left[\frac{1+i}{\alpha}\right]=i^{\frac{b(a-3b)}{2}-\frac{a^2-1}{8}} The law can be stated without using the concept of primary: If λ is odd, let ε(λ) be the unique unit congruent to λ (mod (1 +
i)3); i.e., ε(λ) =
ik ≡ λ (mod 2 + 2
i), where 0 ≤
k ≤ 3. Then for odd and relatively prime α and β, neither one a unit, :\left[\frac{\alpha}{\beta}\right]\left[\frac{\beta}{\alpha}\right]^{-1}= (-1)^{\frac{N\alpha-1}{4}\frac{N\beta-1}{4}}\epsilon(\alpha)^\frac{N\beta-1}{4}\epsilon(\beta)^\frac{N\alpha-1}{4} For odd λ, let \lambda^*=(-1)^\frac{N\lambda-1}{4}\lambda. Then if λ and μ are relatively prime nonunits, Eisenstein proved :\left[\frac{\lambda}{\mu}\right]=\Bigg[\frac{\mu^*}{\lambda}\Bigg]. ==See also==