Cubic reciprocity
Encyclopedia
Cubic reciprocity is a collection of theorems in elementary and algebraic
number theory
that state conditions under which the congruence
x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integer
s, both coprime to 3,
made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.
Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae
(1801). In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818) he said that he was publishing these proofs because their techniques (Gauss's lemma
and Gaussian sums
, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs on biquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.
From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of cubic and biquadratic reciprocity around 1814. Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.
Jacobi published several theorems about cubic residuacity in 1827, but no proofs. In his Königsberg lectures of 1836–37 Jacobi presented proofs. The first published proofs were by Eisenstein (1844).
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 be positive, odd primes.
The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 2 (mod 3) every number is a cubic residue (mod q). Let q = 3n + 2; since 0 = 03 is obviously a cubic residue, assume x is not divisible by q. Then by Fermat's little theorem
,
is a cubic residue (mod q).
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 3).
In this case, p ≡ 1 (mod 3), the nonzero residue classes (mod p) can be divided into three sets, each containing (p−1)/3 numbers. Let e be a cubic nonresidue. The first set is the cubic residues; the second one is e times the numbers in the first set, and the third is e2 times the numbers in the first set. Another way to describe this division is to let e be a primitive root
(mod p); then the first (respectively second, third) set is the numbers whose indices with respect to this root are ≡ 0 (resp. 1, 2) (mod 3). In the vocabulary of group theory
, the first set is a subgroup of index
3 (of the multiplicative group Z/pZ ×), and the other two are its cosets.
Letting m = a + b and n = a − b, we see that this is equivalent to p = m2 − mn + n2 (which equals (n − m)2 − (n − m)n + n2 = m2 + m(n − m) + (n − m)2, so m and n are not determined uniquely). Thus,
and it is a straightforward exercise to show that exactly one of m, n, or m − n is a multiple of 3, so
and this representation is unique up to the signs of L and M.
For relatively-prime integers m and n define the rational cubic residue symbol as
The first two can be restated as
then 
from which 
is an easy deduction.
Let q ≡ p ≡ 1 (mod 6) be positive primes,
and let x be a solution of x2 ≡ −3 (mod q). Then
(The "numerator" in the last expression is an integer (mod q), not a Legendre symbol).
If
then 
, and we have
Along the same lines, von Lienen proved
Let
be primes.
Note that the first condition implies:
The first few examples of this are equivalent to Euler's conjectures:
Martinet proved
Let p ≡ q ≡ 1 (mod 3) be primes,
Then
Sharifi proved
Let p = 1 + 3x + 9x2 be prime. Then
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
In his first monograph on cubic reciprocity Eisenstein developed the theory of the numbers built up from a cube root of unity; they are now called the ring of Eisenstein integers. Eisenstein said (paraphrasing) "to investigate the properties of this ring one need only consult Gauss's work on Z[i] and modify the proofs". This is not surprising since both rings are unique factorization domain
s.
The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers
of the cyclotomic number field
s; the Gaussian and Eisenstein integers are the simplest examples of these.
be a complex cube root of unity. The Eisenstein integers Z[ω] are all numbers of the form a + bω where a and b are ordinary integer
s.
Since ω3 − 1 = (ω − 1)(ω2 + ω + 1) = 0 and ω ≠ 1, we have ω2 = − ω − 1 and ω = − ω2 − 1. Since
and 
where the bar denotes complex conjugation. Also, 
If λ = a + bω and μ = c + dω,
This shows that Z[ω] is closed under addition and multiplication, making it a ring
.
The units are the numbers that divide 1. They are ±1, ±ω, and ±ω2. They are similar to 1 and −1 in the ordinary integers, in that they divide every number. The units are the powers of −ω, a sixth (not just a third) root of unity.
Given a number λ = a + bω, its conjugate means its complex conjugate a + bω2 = (a − b) − bω (not a − bω), and its associates are its six unit multiples:
The norm of λ = a + bω is the product of λ and its conjugate
From the definition, if λ and μ are two Eisenstein integers, Nλμ = Nλ Nμ; in other words, the norm is a completely multiplicative function
. The norm of zero is zero, the norm of any other number is a positive integer. ε is a unit if and only if Nε = 1. Note that the norm is always ≡ 0 or ≡ 1 (mod 3).
Z[ω] is a unique factorization domain
. The primes fall into three classes:
Thus, inert primes are 2, 5, 11, 17, ... and a factorization of the split primes is
The associates and conjugate of a prime are also primes.
Note that the norm of an inert prime q is Nq = q2 ≡ 1 (mod 3).
In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Eisenstein defines a number to be primary if it is ≡ 2 (mod 3). It is straightforward to show that if gcd(Nλ, 3) = 1 then exactly one associate of λ is primary. A disadvantage of this definition is that the product of two primary numbers is the negative of a primary.
Most modern authors say that a number is primary if it is coprime to 3 and congruent to an ordinary integer (mod (1 − ω)2), which is the same as saying it is ≡ ±2 (mod 3). There are two reasons to do this: first, the product of two primaries is a primary, and second, it generalizes to all cyclotomic number fields. Under this definition, if gcd(Nλ, 3) = 1 one of λ, ωλ, or ω2λ is primary. A primary under Eisenstein's definition is primary under the modern one, and if λ is primary under the modern one, either λ or −λ is primary under Eisenstein's. Since −1 is a cube, this does not affect the statement of cubic reciprocity, but it does affect the unique factorization theorem. This article uses the modern definition, so
The product of two primary numbers is primary and the conjugate of a primary number is also primary.
The unique factorization theorem for Z[ω] is: if λ ≠ 0, then
where 0 ≤ μ ≤ 2, ν ≥ 0, each πi is a primary prime, and each αi ≥ 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[ω] 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.
is true in Z[ω]: if α is not divisible by a prime π,
Now assume that Nπ ≠ 3, so that Nπ ≡ 1 (mod 3).
Then
makes sense, and 
for a unique unit ωk.
This unit is called the cubic residue character of α (mod π) and is denoted by
It has formal properties similar to those of the Legendre symbol
.
where the bar denotes complex conjugation.
The cubic character can be extended multiplicatively to composite numbers (coprime to 3) in the "denominator" in the same way the Legendre symbol is generalized into the Jacobi symbol
. Like the Jacobi symbol, if the "denominator" of the cubic character is composite, then if the "numerator" is a cubic residue mod the "denominator" the symbol will equal 1, if the symbol does not equal 1 then the "numerator" is a cubic nonresidue, but the symbol can equal 1 when the "numerator" is a nonresidue:
where 
There are supplementary theorems for the units and the prime 1 − ω:
Let α = a + bω be primary, a = 3m + 1 and b = 3n. (If a ≡ 2 (mod 3) replace α with its associate −α; this will not change the value of the cubic characters.) Then
These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148
Gauss's fifth and sixth proofs of quadratic reciprocity are in
This is in Gauss's Werke, Vol II, pp. 47–64
German translations of all three of the above are the following, which also has the Disquisitiones Arithmeticae
and Gauss's other papers on number theory.
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
that state conditions under which the congruence
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integer
Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
s, both coprime to 3,
- The congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is.
History
Sometime before 1748 EulerLeonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.
Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
(1801). In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818) he said that he was publishing these proofs because their techniques (Gauss's lemma
Gauss's lemma (number theory)
Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity....
and Gaussian sums
Quadratic Gauss sum
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general...
, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs on biquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.
From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of cubic and biquadratic reciprocity around 1814. Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.
Jacobi published several theorems about cubic residuacity in 1827, but no proofs. In his Königsberg lectures of 1836–37 Jacobi presented proofs. The first published proofs were by Eisenstein (1844).
Integers
A cubic residue (mod p) is any number congruent to the third power of an integer (mod p). If x3 ≡ a (mod p) does not have an integer solution, a is a cubic 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 be positive, odd primes.
The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 2 (mod 3) every number is a cubic residue (mod q). Let q = 3n + 2; since 0 = 03 is obviously a cubic residue, assume x is not divisible by q. Then by Fermat's little theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
,
is a cubic residue (mod q).
Therefore, the only interesting case is when the modulus p ≡ 1 (mod 3).
In this case, p ≡ 1 (mod 3), the nonzero residue classes (mod p) can be divided into three sets, each containing (p−1)/3 numbers. Let e be a cubic nonresidue. The first set is the cubic residues; the second one is e times the numbers in the first set, and the third is e2 times the numbers in the first set. Another way to describe this division is to let e be a primitive root
Primitive root modulo n
In modular arithmetic, a branch of number theory, a primitive root modulo n is any number g with the property that any number coprime to n is congruent to a power of g modulo n. In other words, g is a generator of the multiplicative group of integers modulo n...
(mod p); then the first (respectively second, third) set is the numbers whose indices with respect to this root are ≡ 0 (resp. 1, 2) (mod 3). In the vocabulary of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the first set is a subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
3 (of the multiplicative group Z/pZ ×), and the other two are its cosets.
Primes ≡ 1 (mod 3)
A theorem of Fermat states that every prime p ≡ 1 (mod 3) is the sum of a square and three times a square: p = a2 + 3b2 and (except for the signs of a and b) this representation is unique.Letting m = a + b and n = a − b, we see that this is equivalent to p = m2 − mn + n2 (which equals (n − m)2 − (n − m)n + n2 = m2 + m(n − m) + (n − m)2, so m and n are not determined uniquely). Thus,
and it is a straightforward exercise to show that exactly one of m, n, or m − n is a multiple of 3, so
and this representation is unique up to the signs of L and M.For relatively-prime integers m and n define the rational cubic residue symbol as
Euler
Euler's conjectures are based on the representation p = 3a2 + b2. The symbol m|n is read "m divides n" and means there is an a such that n = ma.The first two can be restated as
- Let p ≡ 1 (mod 3) be a positive prime. Then 2 is a cubic residue of p if and only if p = a2 + 27b2.
- Let p ≡ 1 (mod 3) be a positive prime. Then 3 is a cubic residue of p if and only if 4p = a2 + 243b2.
Gauss
Gauss proves that if then from which is an easy deduction.Jacobi
Jacobi stated (without proof)Let q ≡ p ≡ 1 (mod 6) be positive primes,
and let x be a solution of x2 ≡ −3 (mod q). Then(The "numerator" in the last expression is an integer (mod q), not a Legendre symbol).
If
then , and we haveAlong the same lines, von Lienen proved
Other theorems
Emma Lehmer provedLet
be primes.Note that the first condition implies:
- Any number that divides L or M is a cubic residue (mod p).
The first few examples of this are equivalent to Euler's conjectures:
Martinet proved
Let p ≡ q ≡ 1 (mod 3) be primes,
ThenSharifi proved
Let p = 1 + 3x + 9x2 be prime. Then
- Any divisor of x is a cubic residue (mod p).
Background
In his second monograph on biquadratic reciprocity, Gauss says: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
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
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.
In his first monograph on cubic reciprocity Eisenstein developed the theory of the numbers built up from a cube root of unity; they are now called the ring of Eisenstein integers. Eisenstein said (paraphrasing) "to investigate the properties of this ring one need only consult Gauss's work on Z[i] and modify the proofs". This is not surprising since both rings are unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
s.
The "other imaginary quantities" needed for the "theory of residues of higher powers" are the rings of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
of the cyclotomic number field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s; the Gaussian and Eisenstein integers are the simplest examples of these.
Facts and terminology
Let be a complex cube root of unity. The Eisenstein integers Z[ω] are all numbers of the form a + bω where a and b are ordinary integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s.
Since ω3 − 1 = (ω − 1)(ω2 + ω + 1) = 0 and ω ≠ 1, we have ω2 = − ω − 1 and ω = − ω2 − 1. Since
and where the bar denotes complex conjugation. Also, If λ = a + bω and μ = c + dω,
- λ + μ = (a + c) + (b + d)ω and
- λ μ = ac + (ad + bc)ω + bdω2 = (ac − bd) + (ad + bc − bd)ω.
This shows that Z[ω] is closed under addition and multiplication, making it a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
.
The units are the numbers that divide 1. They are ±1, ±ω, and ±ω2. They are similar to 1 and −1 in the ordinary integers, in that they divide every number. The units are the powers of −ω, a sixth (not just a third) root of unity.
Given a number λ = a + bω, its conjugate means its complex conjugate a + bω2 = (a − b) − bω (not a − bω), and its associates are its six unit multiples:
The norm of λ = a + bω is the product of λ and its conjugate
From the definition, if λ and μ are two Eisenstein integers, Nλμ = Nλ Nμ; in other words, the norm is a completely multiplicative functionCompletely multiplicative function
In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. Especially in number theory, a weaker condition is also important, respecting only products of coprime numbers, and such...
. The norm of zero is zero, the norm of any other number is a positive integer. ε is a unit if and only if Nε = 1. Note that the norm is always ≡ 0 or ≡ 1 (mod 3).
Z[ω] is a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
. The primes fall into three classes:
- 3 is a special case: 3 = −ω2(1 − ω)2. It is the only prime in Z divisible by the square of a prime in Z[ω]. In algebraic number theory, 3 is said to ramify in Z[ω].
- Positive primes in Z ≡ 2 (mod 3) are also primes in Z[ω]. In algebraic number theory, these primes are said to remain inert in Z[ω].
- Positive primes in Z ≡ 1 (mod 3) are the product of two conjugate primes in Z[ω].
In algebraic number theory, these primes are said to split in Z[ω].
Thus, inert primes are 2, 5, 11, 17, ... and a factorization of the split primes is
- 7 = (3 + ω) × (2 − ω),
- 13 = (4 + ω) × (3 − ω),
- 19 = (3 − 2ω) × (5 + 2ω),
- 31 = (1 + 6ω) × (−5 − 6ω), ...
The associates and conjugate of a prime are also primes.
Note that the norm of an inert prime q is Nq = q2 ≡ 1 (mod 3).
In order to state the unique factorization theorem, it is necessary to have a way of distinguishing one of the associates of a number. Eisenstein defines a number to be primary if it is ≡ 2 (mod 3). It is straightforward to show that if gcd(Nλ, 3) = 1 then exactly one associate of λ is primary. A disadvantage of this definition is that the product of two primary numbers is the negative of a primary.
Most modern authors say that a number is primary if it is coprime to 3 and congruent to an ordinary integer (mod (1 − ω)2), which is the same as saying it is ≡ ±2 (mod 3). There are two reasons to do this: first, the product of two primaries is a primary, and second, it generalizes to all cyclotomic number fields. Under this definition, if gcd(Nλ, 3) = 1 one of λ, ωλ, or ω2λ is primary. A primary under Eisenstein's definition is primary under the modern one, and if λ is primary under the modern one, either λ or −λ is primary under Eisenstein's. Since −1 is a cube, this does not affect the statement of cubic reciprocity, but it does affect the unique factorization theorem. This article uses the modern definition, so
The product of two primary numbers is primary and the conjugate of a primary number is also primary.
The unique factorization theorem for Z[ω] is: if λ ≠ 0, then
where 0 ≤ μ ≤ 2, ν ≥ 0, each πi is a primary prime, and each αi ≥ 1, and this representation is unique, up to the order of the factors.
The notions of congruence
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
and greatest common divisor
Greatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
are defined the same way in Z[ω] 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.
Cubic residue character
An analogue of Fermat's little theoremFermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
is true in Z[ω]: if α is not divisible by a prime π,
Now assume that Nπ ≠ 3, so that Nπ ≡ 1 (mod 3).
Then
makes sense, and for a unique unit ωk.This unit is called the cubic residue character of α (mod π) and is denoted by
It has formal properties similar to those of the Legendre symbol
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
.
- The congruence
is solvable in Z[ω] if and only if 
where the bar denotes complex conjugation.- if π and θ are associates,
- if α ≡ β (mod π),
The cubic character can be extended multiplicatively to composite numbers (coprime to 3) in the "denominator" in the same way the Legendre symbol is generalized into the Jacobi symbol
Jacobi symbol
The Jacobi symbol is a generalization of the Legendre symbol. Introduced by Jacobi in 1837, it is of theoretical interest in modular arithmetic and other branches of number theory, but its main use is in computational number theory, especially primality testing and integer factorization; these in...
. Like the Jacobi symbol, if the "denominator" of the cubic character is composite, then if the "numerator" is a cubic residue mod the "denominator" the symbol will equal 1, if the symbol does not equal 1 then the "numerator" is a cubic nonresidue, but the symbol can equal 1 when the "numerator" is a nonresidue:
where - If a and b are ordinary integers, gcd(a, b) = gcd(b, 3) = 1, then
Statement of the theorem
Let α and β be primary. ThenThere are supplementary theorems for the units and the prime 1 − ω:
Let α = a + bω be primary, a = 3m + 1 and b = 3n. (If a ≡ 2 (mod 3) replace α with its associate −α; this will not change the value of the cubic characters.) Then
Euler
This was actually written 1748–1750, but was only published posthumously; It is in Vol V, pp. 182–283 ofGauss
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". Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n".These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148
Gauss's fifth and sixth proofs of quadratic reciprocity are in
This is in Gauss's Werke, Vol II, pp. 47–64
German translations of all three of the above are the following, which also has the Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
and Gauss's other papers on number theory.