# Quadratic reciprocity

Quadratic reciprocity Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125–146 and 262 of Disquisitiones Arithmeticae in 1801.

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: Law of quadratic reciprocity — Let p and q be distinct odd prime numbers, and define the Legendre symbol as: {displaystyle left({frac {q}{p}}right)={begin{cases}1&{text{if }}n^{2}equiv q{bmod {p}}{text{ for some integer }}n\-1&{text{otherwise}}end{cases}}} Then: {displaystyle left({frac {p}{q}}right)left({frac {q}{p}}right)=(-1)^{{frac {p-1}{2}}{frac {q-1}{2}}}.} This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form {displaystyle x^{2}equiv a{bmod {p}}} for an odd prime {displaystyle p} ; that is, to determine the "perfect squares" modulo {displaystyle p} . However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required. For example, in the case {displaystyle pequiv 3{bmod {4}}} using Euler's criterion one can give an explicit formula for the "square roots" modulo {displaystyle p} of a quadratic residue {displaystyle a} , namely, {displaystyle pm a^{frac {p+1}{4}}} indeed, {displaystyle left(pm a^{frac {p+1}{4}}right)^{2}=a^{frac {p+1}{2}}=acdot a^{frac {p-1}{2}}equiv aleft({frac {a}{p}}right)=a{bmod {p}}.} This formula only works if it is known in advance that {displaystyle a} is a quadratic residue, which can be checked using the law of quadratic reciprocity.

The quadratic reciprocity theorem was conjectured by Euler and Legendre and first proved by Gauss,[1] who referred to it as the "fundamental theorem" in his Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type. (Art. 151) Privately, Gauss referred to it as the "golden theorem".[2] He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs.[3] The shortest known proof is included below, together with short proofs of the law's supplements (the Legendre symbols of −1 and 2).

Generalizing the reciprocity law to higher powers has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program.

Contents 1 Motivating examples 1.1 Factoring n2 − 5 1.2 Patterns among quadratic residues 1.3 Legendre's version 2 Supplements to Quadratic Reciprocity 2.1 q = ±1 and the first supplement 2.2 q = ±2 and the second supplement 2.3 q = ±3 2.4 q = ±5 2.5 Higher q 3 Statement of the theorem 4 Proof 4.1 Proofs of the supplements 5 History and alternative statements 5.1 Fermat 5.2 Euler 5.3 Legendre and his symbol 5.3.1 Legendre's version of quadratic reciprocity 5.3.2 The supplementary laws using Legendre symbols 5.4 Gauss 5.5 Other statements 5.6 Jacobi symbol 5.7 Hilbert symbol 6 Connection with cyclotomic fields 7 Other rings 7.1 Gaussian integers 7.2 Eisenstein integers 7.3 Imaginary quadratic fields 7.4 Polynomials over a finite field 8 Higher powers 9 See also 10 Notes 11 References 12 External links Motivating examples Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers. In this section, we give examples which lead to the general case.

Factoring n2 − 5 Consider the polynomial {displaystyle f(n)=n^{2}-5} and its values for {displaystyle nin mathbb {N} .} The prime factorizations of these values are given as follows: n {displaystyle f(n)} n {displaystyle f(n)} n {displaystyle f(n)} 1 −4 −22 16 251 251 31 956 22⋅239 2 −1 −1 17 284 22⋅71 32 1019 1019 3 4 22 18 319 11⋅29 33 1084 22⋅271 4 11 11 19 356 22⋅89 34 1151 1151 5 20 22⋅5 20 395 5⋅79 35 1220 22⋅5⋅61 6 31 31 21 436 22⋅109 36 1291 1291 7 44 22⋅11 22 479 479 37 1364 22⋅11⋅31 8 59 59 23 524 22⋅131 38 1439 1439 9 76 22⋅19 24 571 571 39 1516 22⋅379 10 95 5⋅19 25 620 22⋅5⋅31 40 1595 5⋅11⋅29 11 116 22⋅29 26 671 11⋅61 41 1676 22⋅419 12 139 139 27 724 22⋅181 42 1759 1759 13 164 22⋅41 28 779 19⋅41 43 1844 22⋅461 14 191 191 29 836 22⋅11⋅19 44 1931 1931 15 220 22⋅5⋅11 30 895 5⋅179 45 2020 22⋅5⋅101 The prime factors {displaystyle p} dividing {displaystyle f(n)} are {displaystyle p=2,5} , and every prime whose final digit is {displaystyle 1} or {displaystyle 9} ; no primes ending in {displaystyle 3} or {displaystyle 7} ever appear. Now, {displaystyle p} is a prime factor of some {displaystyle n^{2}-5} whenever {displaystyle n^{2}-5equiv 0{bmod {p}}} , i.e. whenever {displaystyle n^{2}equiv 5{bmod {p}},} i.e. whenever 5 is a quadratic residue modulo {displaystyle p} . This happens for {displaystyle p=2,5} and those primes with {displaystyle pequiv 1,4{bmod {5}},} and the latter numbers {displaystyle 1=(pm 1)^{2}} and {displaystyle 4=(pm 2)^{2}} are precisely the quadratic residues modulo {displaystyle 5} . Therefore, except for {displaystyle p=2,5} , we have that {displaystyle 5} is a quadratic residue modulo {displaystyle p} iff {displaystyle p} is a quadratic residue modulo {displaystyle 5} .

The law of quadratic reciprocity gives a similar characterization of prime divisors of {displaystyle f(n)=n^{2}-q} for any prime q, which leads to a characterization for any integer {displaystyle q} .

Patterns among quadratic residues Let p be an odd prime. A number modulo p is a quadratic residue whenever it is congruent to a square (mod p); otherwise it is a quadratic non-residue. ("Quadratic" can be dropped if it is clear from the context.) Here we exclude zero as a special case. Then as a consequence of the fact that the multiplicative group of a finite field of order p is cyclic of order p-1, the following statements hold: There are an equal number of quadratic residues and non-residues; and The product of two quadratic residues is a residue, the product of a residue and a non-residue is a non-residue, and the product of two non-residues is a residue.

For the avoidance of doubt, these statements do not hold if the modulus is not prime. For example, there are only 3 quadratic residues (1, 4 and 9) in the multiplicative group modulo 15. Moreover, although 7 and 8 are quadratic non-residues, their product 7x8 = 11 is also a quadratic non-residue, in contrast to the prime case.

Quadratic residues are entries in the following table: Squares mod primes n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 n2 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 mod 3 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 mod 5 1 4 4 1 0 1 4 4 1 0 1 4 4 1 0 1 4 4 1 0 1 4 4 1 0 mod 7 1 4 2 2 4 1 0 1 4 2 2 4 1 0 1 4 2 2 4 1 0 1 4 2 2 mod 11 1 4 9 5 3 3 5 9 4 1 0 1 4 9 5 3 3 5 9 4 1 0 1 4 9 mod 13 1 4 9 3 12 10 10 12 3 9 4 1 0 1 4 9 3 12 10 10 12 3 9 4 1 mod 17 1 4 9 16 8 2 15 13 13 15 2 8 16 9 4 1 0 1 4 9 16 8 2 15 13 mod 19 1 4 9 16 6 17 11 7 5 5 7 11 17 6 16 9 4 1 0 1 4 9 16 6 17 mod 23 1 4 9 16 2 13 3 18 12 8 6 6 8 12 18 3 13 2 16 9 4 1 0 1 4 mod 29 1 4 9 16 25 7 20 6 23 13 5 28 24 22 22 24 28 5 13 23 6 20 7 25 16 mod 31 1 4 9 16 25 5 18 2 19 7 28 20 14 10 8 8 10 14 20 28 7 19 2 18 5 mod 37 1 4 9 16 25 36 12 27 7 26 10 33 21 11 3 34 30 28 28 30 34 3 11 21 33 mod 41 1 4 9 16 25 36 8 23 40 18 39 21 5 32 20 10 2 37 33 31 31 33 37 2 10 mod 43 1 4 9 16 25 36 6 21 38 14 35 15 40 24 10 41 31 23 17 13 11 11 13 17 23 mod 47 1 4 9 16 25 36 2 17 34 6 27 3 28 8 37 21 7 42 32 24 18 14 12 12 14 This table is complete for odd primes less than 50. To check whether a number m is a quadratic residue mod one of these primes p, find a ≡ m (mod p) and 0 ≤ a < p. If a is in row p, then m is a residue (mod p); if a is not in row p of the table, then m is a nonresidue (mod p). The quadratic reciprocity law is the statement that certain patterns found in the table are true in general. Legendre's version Another way to organize the data is to see which primes are residues mod which other primes, as illustrated in the following table. The entry in row p column q is R if q is a quadratic residue (mod p); if it is a nonresidue the entry is N. If the row, or the column, or both, are ≡ 1 (mod 4) the entry is blue or green; if both row and column are ≡ 3 (mod 4), it is yellow or orange. The blue and green entries are symmetric around the diagonal: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is R (resp N). The yellow and orange ones, on the other hand, are antisymmetric: The entry for row p, column q is R (resp N) if and only if the entry at row q, column p, is N (resp R). The reciprocity law states that these patterns hold for all p and q. Legend R q is a residue (mod p) q ≡ 1 (mod 4) or p ≡ 1 (mod 4) (or both) N q is a nonresidue (mod p) R q is a residue (mod p) both q ≡ 3 (mod 4) and p ≡ 3 (mod 4) N q is a nonresidue (mod p) q 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 p 3 N R N R N R N N R R N R N N N R R N R R N N R 5 N N R N N R N R R N R N N N R R N R N R N R N 7 N N R N N N R R N R N R N R N N R R N R N N N 11 R R N N N N R N R R N N R R R N R R N N N R R 13 R N N N R N R R N N N R N R N R N N N R N N N 17 N N N N R R N N N N N R R R R N R N N N R R N 19 N R R R N R R N N N N R R N N R N N R N R N N 23 R N N N R N N R R N R N R N R N N R R N N N N 29 N R R N R N N R N N N N N R R N R R N N R N N 31 N R R N N N R N N N R N R N R N R R N N N N R 37 R N R R N N N N N N R N R R N N R R R N R N N 41 N R N N N N N R N R R R N N R R N N R N R N N 43 N N N R R R N R N R N R R R R N R N N R R N R 47 R N R N N R N N N N R N N R R R N R N R R R R 53 N N R R R R N N R N R N R R R N N N N N N R R 59 R R R N N R R N R N N R N N R N N R N R N N N 61 R R N N R N R N N N N R N R N N N N R N R N R 67 N N N N N R R R R N R N N R N R N R R N R R N 71 R R N N N N R N R N R N R N N N N N R R R R N 73 R N N N N N R R N N R R N N N N R R R R N R R 79 N R N R R N R R N R N N N N N N N R N R R R R 83 R N R R N R N R R R R R N N N R R N N N N N N 89 N R N R N R N N N N N N N R R N N R R R R N R 97 R N N R N N N N N R N N R R R N R N N R R N R Supplements to Quadratic Reciprocity The supplements provide solutions to specific cases of quadratic reciprocity. They are often quoted as partial results, without having to resort to the complete theorem. q = ±1 and the first supplement Trivially 1 is a quadratic residue for all primes. The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47. The former set of primes are all congruent to 1 modulo 4, and the latter are congruent to 3 modulo 4. First Supplement to Quadratic Reciprocity. The congruence {displaystyle x^{2}equiv -1{bmod {p}}} is solvable if and only if {displaystyle p} is congruent to 1 modulo 4. q = ±2 and the second supplement Examining the table, we find 2 in rows 7, 17, 23, 31, 41, and 47, but not in rows 3, 5, 11, 13, 19, 29, 37, or 43. The former primes are all ≡ ±1 (mod 8), and the latter are all ≡ ±3 (mod 8). This leads to Second Supplement to Quadratic Reciprocity. The congruence {displaystyle x^{2}equiv 2{bmod {p}}} is solvable if and only if {displaystyle p} is congruent to ±1 modulo 8. −2 is in rows 3, 11, 17, 19, 41, 43, but not in rows 5, 7, 13, 23, 29, 31, 37, or 47. The former are ≡ 1 or ≡ 3 (mod 8), and the latter are ≡ 5, 7 (mod 8). q = ±3 3 is in rows 11, 13, 23, 37, and 47, but not in rows 5, 7, 17, 19, 29, 31, 41, or 43. The former are ≡ ±1 (mod 12) and the latter are all ≡ ±5 (mod 12). −3 is in rows 7, 13, 19, 31, 37, and 43 but not in rows 5, 11, 17, 23, 29, 41, or 47. The former are ≡ 1 (mod 3) and the latter ≡ 2 (mod 3). Since the only residue (mod 3) is 1, we see that −3 is a quadratic residue modulo every prime which is a residue modulo 3. q = ±5 5 is in rows 11, 19, 29, 31, and 41 but not in rows 3, 7, 13, 17, 23, 37, 43, or 47. The former are ≡ ±1 (mod 5) and the latter are ≡ ±2 (mod 5). Since the only residues (mod 5) are ±1, we see that 5 is a quadratic residue modulo every prime which is a residue modulo 5. −5 is in rows 3, 7, 23, 29, 41, 43, and 47 but not in rows 11, 13, 17, 19, 31, or 37. The former are ≡ 1, 3, 7, 9 (mod 20) and the latter are ≡ 11, 13, 17, 19 (mod 20). Higher q The observations about −3 and 5 continue to hold: −7 is a residue modulo p if and only if p is a residue modulo 7, −11 is a residue modulo p if and only if p is a residue modulo 11, 13 is a residue (mod p) if and only if p is a residue modulo 13, etc. The more complicated-looking rules for the quadratic characters of 3 and −5, which depend upon congruences modulo 12 and 20 respectively, are simply the ones for −3 and 5 working with the first supplement. Example. For −5 to be a residue (mod p), either both 5 and −1 have to be residues (mod p) or they both have to be non-residues: i.e., p ≡ ±1 (mod 5) and p ≡ 1 (mod 4) or p ≡ ±2 (mod 5) and p ≡ 3 (mod 4). Using the Chinese remainder theorem these are equivalent to p ≡ 1, 9 (mod 20) or p ≡ 3, 7 (mod 20). The generalization of the rules for −3 and 5 is Gauss's statement of quadratic reciprocity. Statement of the theorem Quadratic Reciprocity (Gauss's statement). If {displaystyle qequiv 1{bmod {4}}} , then the congruence {displaystyle x^{2}equiv p{bmod {q}}} is solvable if and only if {displaystyle x^{2}equiv q{bmod {p}}} is solvable. If {displaystyle qequiv 3{bmod {4}}} and {displaystyle pequiv 3{bmod {4}}} , then the congruence {displaystyle x^{2}equiv p{bmod {q}}} is solvable if and only if {displaystyle x^{2}equiv -q{bmod {p}}} is solvable. Quadratic Reciprocity (combined statement). Define {displaystyle q^{*}=(-1)^{frac {q-1}{2}}q} . Then the congruence {displaystyle x^{2}equiv p{bmod {q}}} is solvable if and only if {displaystyle x^{2}equiv q^{*}{bmod {p}}} is solvable. Quadratic Reciprocity (Legendre's statement). If p or q are congruent to 1 modulo 4, then: {displaystyle x^{2}equiv q{bmod {p}}} is solvable if and only if {displaystyle x^{2}equiv p{bmod {q}}} is solvable. If p and q are congruent to 3 modulo 4, then: {displaystyle x^{2}equiv q{bmod {p}}} is solvable if and only if {displaystyle x^{2}equiv p{bmod {q}}} is not solvable. The last is immediately equivalent to the modern form stated in the introduction above. It is a simple exercise to prove that Legendre's and Gauss's statements are equivalent – it requires no more than the first supplement and the facts about multiplying residues and nonresidues. Proof Main article: Proofs of quadratic reciprocity Apparently, the shortest known proof yet was published by B. Veklych in the American Mathematical Monthly.[4] Proofs of the supplements The value of the Legendre symbol of {displaystyle -1} (used in the proof above) follows directly from Euler's criterion: {displaystyle left({frac {-1}{p}}right)equiv (-1)^{frac {p-1}{2}}{bmod {p}}} by Euler's criterion, but both sides of this congruence are numbers of the form {displaystyle pm 1} , so they must be equal. Whether {displaystyle 2} is a quadratic residue can be concluded if we know the number of solutions of the equation {displaystyle x^{2}+y^{2}=2} with {displaystyle x,yin mathbb {Z} _{p},} which can be solved by standard methods. Namely, all its solutions where {displaystyle xyneq 0,xneq pm y} can be grouped into octuplets of the form {displaystyle (pm x,pm y),(pm y,pm x)} , and what is left are four solutions of the form {displaystyle (pm 1,pm 1)} and possibly four additional solutions where {displaystyle x^{2}=2,y=0} and {displaystyle x=0,y^{2}=2} , which exist precisely if {displaystyle 2} is a quadratic residue. That is, {displaystyle 2} is a quadratic residue precisely if the number of solutions of this equation is divisible by {displaystyle 8} . And this equation can be solved in just the same way here as over the rational numbers: substitute {displaystyle x=a+1,y=at+1} , where we demand that {displaystyle aneq 0} (leaving out the two solutions {displaystyle (1,pm 1)} ), then the original equation transforms into {displaystyle a=-{frac {2(t+1)}{(t^{2}+1)}}.} Here {displaystyle t} can have any value that does not make the denominator zero – for which there are {displaystyle 1+left({frac {-1}{p}}right)} possibilities (i.e. {displaystyle 2} if {displaystyle -1} is a residue, {displaystyle 0} if not) – and also does not make {displaystyle a} zero, which excludes one more option, {displaystyle t=-1} . Thus there are {displaystyle p-left(1+left({frac {-1}{p}}right)right)-1} possibilities for {displaystyle t} , and so together with the two excluded solutions there are overall {displaystyle p-left({frac {-1}{p}}right)} solutions of the original equation. Therefore, {displaystyle 2} is a residue modulo {displaystyle p} if and only if {displaystyle 8} divides {displaystyle p-(-1)^{frac {p-1}{2}}} . This is a reformulation of the condition stated above. History and alternative statements The theorem was formulated in many ways before its modern form: Euler and Legendre did not have Gauss's congruence notation, nor did Gauss have the Legendre symbol. In this article p and q always refer to distinct positive odd primes, and x and y to unspecified integers. Fermat Fermat proved[5] (or claimed to have proved)[6] a number of theorems about expressing a prime by a quadratic form: {displaystyle {begin{aligned}p=x^{2}+y^{2}qquad &Longleftrightarrow qquad p=2quad {text{ or }}quad pequiv 1{bmod {4}}\p=x^{2}+2y^{2}qquad &Longleftrightarrow qquad p=2quad {text{ or }}quad pequiv 1,3{bmod {8}}\p=x^{2}+3y^{2}qquad &Longleftrightarrow qquad p=3quad {text{ or }}quad pequiv 1{bmod {3}}\end{aligned}}} He did not state the law of quadratic reciprocity, although the cases −1, ±2, and ±3 are easy deductions from these and other of his theorems. He also claimed to have a proof that if the prime number p ends with 7, (in base 10) and the prime number q ends in 3, and p ≡ q ≡ 3 (mod 4), then {displaystyle pq=x^{2}+5y^{2}.} Euler conjectured, and Lagrange proved, that[7] {displaystyle {begin{aligned}p&equiv 1,9{bmod {20}}quad Longrightarrow quad p=x^{2}+5y^{2}\p,q&equiv 3,7{bmod {20}}quad Longrightarrow quad pq=x^{2}+5y^{2}end{aligned}}} Proving these and other statements of Fermat was one of the things that led mathematicians to the reciprocity theorem. Euler Translated into modern notation, Euler stated [8] that for distinct odd primes p and q: If q ≡ 1 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b such that p ≡ b2 (mod q). If q ≡ 3 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b which is odd and not divisible by q such that p ≡ ±b2 (mod 4q). This is equivalent to quadratic reciprocity. He could not prove it, but he did prove the second supplement.[9] Legendre and his symbol Fermat proved that if p is a prime number and a is an integer, {displaystyle a^{p}equiv a{bmod {p}}.} Thus if p does not divide a, using the non-obvious fact (see for example Ireland and Rosen below) that the residues modulo p form a field and therefore in particular the multiplicative group is cyclic, hence there can be at most two solutions to a quadratic equation: {displaystyle a^{frac {p-1}{2}}equiv pm 1{bmod {p}}.} Legendre[10] lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity: Theorem When it follows that I {displaystyle b^{frac {a-1}{2}}equiv 1{bmod {a}}} {displaystyle a^{frac {b-1}{2}}equiv 1{bmod {b}}} II {displaystyle a^{frac {b-1}{2}}equiv -1{bmod {b}}} {displaystyle b^{frac {a-1}{2}}equiv -1{bmod {a}}} III {displaystyle a^{frac {A-1}{2}}equiv 1{bmod {A}}} {displaystyle A^{frac {a-1}{2}}equiv 1{bmod {a}}} IV {displaystyle a^{frac {A-1}{2}}equiv -1{bmod {A}}} {displaystyle A^{frac {a-1}{2}}equiv -1{bmod {a}}} V {displaystyle a^{frac {b-1}{2}}equiv 1{bmod {b}}} {displaystyle b^{frac {a-1}{2}}equiv 1{bmod {a}}} VI {displaystyle b^{frac {a-1}{2}}equiv -1{bmod {a}}} {displaystyle a^{frac {b-1}{2}}equiv -1{bmod {b}}} VII {displaystyle b^{frac {B-1}{2}}equiv 1{bmod {B}}} {displaystyle B^{frac {b-1}{2}}equiv -1{bmod {b}}} VIII {displaystyle b^{frac {B-1}{2}}equiv -1{bmod {B}}} {displaystyle B^{frac {b-1}{2}}equiv 1{bmod {b}}} He says that since expressions of the form {displaystyle N^{frac {c-1}{2}}{bmod {c}},qquad gcd(N,c)=1} will come up so often he will abbreviate them as: {displaystyle left({frac {N}{c}}right)equiv N^{frac {c-1}{2}}{bmod {c}}=pm 1.} This is now known as the Legendre symbol, and an equivalent[11][12] definition is used today: for all integers a and all odd primes p {displaystyle left({frac {a}{p}}right)={begin{cases}0&aequiv 0{bmod {p}}\1&anot equiv 0{bmod {p}}{text{ and }}exists x:aequiv x^{2}{bmod {p}}\-1&anot equiv 0{bmod {p}}{text{ and there is no such }}x.end{cases}}} Legendre's version of quadratic reciprocity {displaystyle left({frac {p}{q}}right)={begin{cases}left({tfrac {q}{p}}right)&pequiv 1{bmod {4}}quad {text{ or }}quad qequiv 1{bmod {4}}\-left({tfrac {q}{p}}right)&pequiv 3{bmod {4}}quad {text{ and }}quad qequiv 3{bmod {4}}end{cases}}} He notes that these can be combined: {displaystyle left({frac {p}{q}}right)left({frac {q}{p}}right)=(-1)^{{frac {p-1}{2}}{frac {q-1}{2}}}.} A number of proofs, especially those based on Gauss's Lemma,[13] explicitly calculate this formula. The supplementary laws using Legendre symbols {displaystyle {begin{aligned}left({frac {-1}{p}}right)&=(-1)^{frac {p-1}{2}}={begin{cases}1&pequiv 1{bmod {4}}\-1&pequiv 3{bmod {4}}end{cases}}\left({frac {2}{p}}right)&=(-1)^{frac {p^{2}-1}{8}}={begin{cases}1&pequiv 1,7{bmod {8}}\-1&pequiv 3,5{bmod {8}}end{cases}}end{aligned}}} From these two supplements, we can obtain a third reciprocity law for the quadratic character -2 as follows: For -2 to be a quadratic residue, either -1 or 2 are both quadratic residues, or both non-residues : {displaystyle {bmod {p}}} . So either : {displaystyle {frac {p-1}{2}}{text{ or }}{frac {p^{2}-1}{8}}} are both even, or they are both odd. The sum of these two expressions is {displaystyle {frac {p^{2}+4p-5}{8}}} which is an integer. Therefore, {displaystyle {begin{aligned}left({frac {-2}{p}}right)&=(-1)^{frac {p^{2}+4p-5}{8}}={begin{cases}1&pequiv 1,3{bmod {8}}\-1&pequiv 5,7{bmod {8}}end{cases}}end{aligned}}} Legendre's attempt to prove reciprocity is based on a theorem of his: Legendre's Theorem. Let a, b and c be integers where any pair of the three are relatively prime. Moreover assume that at least one of ab, bc or ca is negative (i.e. they don't all have the same sign). If {displaystyle {begin{aligned}u^{2}&equiv -bc{bmod {a}}\v^{2}&equiv -ca{bmod {b}}\w^{2}&equiv -ab{bmod {c}}end{aligned}}} are solvable then the following equation has a nontrivial solution in integers: {displaystyle ax^{2}+by^{2}+cz^{2}=0.} Example. Theorem I is handled by letting a ≡ 1 and b ≡ 3 (mod 4) be primes and assuming that {displaystyle left({tfrac {b}{a}}right)=1} and, contrary the theorem, that {displaystyle left({tfrac {a}{b}}right)=-1.} Then {displaystyle x^{2}+ay^{2}-bz^{2}=0} has a solution, and taking congruences (mod 4) leads to a contradiction. This technique doesn't work for Theorem VIII. Let b ≡ B ≡ 3 (mod 4), and assume {displaystyle left({frac {B}{b}}right)=left({frac {b}{B}}right)=-1.} Then if there is another prime p ≡ 1 (mod 4) such that {displaystyle left({frac {p}{b}}right)=left({frac {p}{B}}right)=-1,} the solvability of {displaystyle Bx^{2}+by^{2}-pz^{2}=0} leads to a contradiction (mod 4). But Legendre was unable to prove there has to be such a prime p; he was later able to show that all that is required is: Legendre's Lemma. If p is a prime that is congruent to 1 modulo 4 then there exists an odd prime q such that {displaystyle left({tfrac {p}{q}}right)=-1.} but he couldn't prove that either. Hilbert symbol (below) discusses how techniques based on the existence of solutions to {displaystyle ax^{2}+by^{2}+cz^{2}=0} can be made to work. Gauss Part of Article 131 in the first edition (1801) of the Disquisitiones, listing the 8 cases of quadratic reciprocity Gauss first proves[14] the supplementary laws. He sets[15] the basis for induction by proving the theorem for ±3 and ±5. Noting[16] that it is easier to state for −3 and +5 than it is for +3 or −5, he states[17] the general theorem in the form: If p is a prime of the form 4n + 1 then p, but if p is of the form 4n + 3 then −p, is a quadratic residue (resp. nonresidue) of every prime, which, with a positive sign, is a residue (resp. nonresidue) of p. In the next sentence, he christens it the "fundamental theorem" (Gauss never used the word "reciprocity"). Introducing the notation a R b (resp. a N b) to mean a is a quadratic residue (resp. nonresidue) (mod b), and letting a, a′, etc. represent positive primes ≡ 1 (mod 4) and b, b′, etc. positive primes ≡ 3 (mod 4), he breaks it out into the same 8 cases as Legendre: Case If Then 1) ±a R a′ ±a′ R a 2) ±a N a′ ±a′ N a 3) +a R b −a N b ±b R a 4) +a N b −a R b ±b N a 5) ±b R a +a R b −a N b 6) ±b N a +a N b −a R b 7) +b R b′ −b N b′ −b′ N b +b′ R b 8) −b N b′ +b R b′ +b′ R b −b′ N b In the next Article he generalizes this to what are basically the rules for the Jacobi symbol (below). Letting A, A′, etc. represent any (prime or composite) positive numbers ≡ 1 (mod 4) and B, B′, etc. positive numbers ≡ 3 (mod 4): Case If Then 9) ±a R A ±A R a 10) ±b R A +A R b −A N b 11) +a R B ±B R a 12) −a R B ±B N a 13) +b R B −B N b +N R b 14) −b R B +B R b −B N b All of these cases take the form "if a prime is a residue (mod a composite), then the composite is a residue or nonresidue (mod the prime), depending on the congruences (mod 4)". He proves that these follow from cases 1) - 8). Gauss needed, and was able to prove,[18] a lemma similar to the one Legendre needed: Gauss's Lemma. If p is a prime congruent to 1 modulo 8 then there exists an odd prime q such that: {displaystyle q<2{sqrt {p}}+1quad {text{and}}quad left({frac {p}{q}}right)=-1.} The proof of quadratic reciprocity uses complete induction. Gauss's Version in Legendre Symbols. {displaystyle left({frac {p}{q}}right)={begin{cases}left({frac {q}{p}}right)&qequiv 1{bmod {4}}\left({frac {-q}{p}}right)&qequiv 3{bmod {4}}end{cases}}} These can be combined: Gauss's Combined Version in Legendre Symbols. Let {displaystyle q^{*}=(-1)^{frac {q-1}{2}}q.} In other words: {displaystyle |q^{*}|=|q|quad {text{and}}quad q^{*}equiv 1{bmod {4}}.} Then: {displaystyle left({frac {p}{q}}right)=left({frac {q^{*}}{p}}right).} A number of proofs of the theorem, especially those based on Gauss sums derive this formula.[19] or the splitting of primes in algebraic number fields,[20] Other statements The statements in this section are equivalent to quadratic reciprocity: if, for example, Euler's version is assumed, the Legendre-Gauss version can be deduced from it, and vice versa. Euler's Formulation of Quadratic Reciprocity.[21] If {displaystyle pequiv pm q{bmod {4a}}} then {displaystyle left({tfrac {a}{p}}right)=left({tfrac {a}{q}}right).} This can be proven using Gauss's lemma. Quadratic Reciprocity (Gauss; Fourth Proof).[22] Let a, b, c, ... be unequal positive odd primes, whose product is n, and let m be the number of them that are ≡ 3 (mod 4); check whether n/a is a residue of a, whether n/b is a residue of b, .... The number of nonresidues found will be even when m ≡ 0, 1 (mod 4), and it will be odd if m ≡ 2, 3 (mod 4). Gauss's fourth proof consists of proving this theorem (by comparing two formulas for the value of Gauss sums) and then restricting it to two primes. He then gives an example: Let a = 3, b = 5, c = 7, and d = 11. Three of these, 3, 7, and 11 ≡ 3 (mod 4), so m ≡ 3 (mod 4). 5×7×11 R 3; 3×7×11 R 5; 3×5×11 R 7; and 3×5×7 N 11, so there are an odd number of nonresidues. Eisenstein's Formulation of Quadratic Reciprocity.[23] Assume {displaystyle pneq q,quad p'neq q',quad pequiv p'{bmod {4}},quad qequiv q'{bmod {4}}.} Then {displaystyle left({frac {p}{q}}right)left({frac {q}{p}}right)=left({frac {p'}{q'}}right)left({frac {q'}{p'}}right).} Mordell's Formulation of Quadratic Reciprocity.[24] Let a, b and c be integers. For every prime, p, dividing abc if the congruence {displaystyle ax^{2}+by^{2}+cz^{2}equiv 0{bmod {tfrac {4abc}{p}}}} has a nontrivial solution, then so does: {displaystyle ax^{2}+by^{2}+cz^{2}equiv 0{bmod {4abc}}.} Zeta function formulation As mentioned in the article on Dedekind zeta functions, quadratic reciprocity is equivalent to the zeta function of a quadratic field being the product of the Riemann zeta function and a certain Dirichlet L-function Jacobi symbol Main article: Jacobi symbol The Jacobi symbol is a generalization of the Legendre symbol; the main difference is that the bottom number has to be positive and odd, but does not have to be prime. If it is prime, the two symbols agree. It obeys the same rules of manipulation as the Legendre symbol. In particular {displaystyle {begin{aligned}left({frac {-1}{n}}right)=(-1)^{frac {n-1}{2}}&={begin{cases}1&nequiv 1{bmod {4}}\-1&nequiv 3{bmod {4}}end{cases}}\left({frac {2}{n}}right)=(-1)^{frac {n^{2}-1}{8}}&={begin{cases}1&nequiv 1,7{bmod {8}}\-1&nequiv 3,5{bmod {8}}end{cases}}\left({frac {-2}{n}}right)=(-1)^{frac {n^{2}+4n-5}{8}}&={begin{cases}1&nequiv 1,3{bmod {8}}\-1&nequiv 5,7{bmod {8}}end{cases}}end{aligned}}} and if both numbers are positive and odd (this is sometimes called "Jacobi's reciprocity law"): {displaystyle left({frac {m}{n}}right)=(-1)^{frac {(m-1)(n-1)}{4}}left({frac {n}{m}}right).} However, if the Jacobi symbol is 1 but the denominator is not a prime, it does not necessarily follow that the numerator is a quadratic residue of the denominator. Gauss's cases 9) - 14) above can be expressed in terms of Jacobi symbols: {displaystyle left({frac {M}{p}}right)=(-1)^{frac {(p-1)(M-1)}{4}}left({frac {p}{M}}right),} and since p is prime the left hand side is a Legendre symbol, and we know whether M is a residue modulo p or not. The formulas listed in the preceding section are true for Jacobi symbols as long as the symbols are defined. Euler's formula may be written {displaystyle left({frac {a}{m}}right)=left({frac {a}{mpm 4an}}right),qquad nin mathbb {Z} ,mpm 4an>0.} Example.

{displaystyle left({frac {2}{7}}right)=left({frac {2}{15}}right)=left({frac {2}{23}}right)=left({frac {2}{31}}right)=cdots =1.} 2 is a residue modulo the primes 7, 23 and 31: {displaystyle 3^{2}equiv 2{bmod {7}},quad 5^{2}equiv 2{bmod {23}},quad 8^{2}equiv 2{bmod {31}}.} But 2 is not a quadratic residue modulo 5, so it can't be one modulo 15. This is related to the problem Legendre had: if {displaystyle left({tfrac {a}{m}}right)=-1,} then a is a non-residue modulo every prime in the arithmetic progression m + 4a, m + 8a, ..., if there are any primes in this series, but that wasn't proved until decades after Legendre.[25] Eisenstein's formula requires relative primality conditions (which are true if the numbers are prime) Let {displaystyle a,b,a',b'} be positive odd integers such that: {displaystyle {begin{aligned}gcd &(a,b)=gcd(a',b')=1\&aequiv a'{bmod {4}}\&bequiv b'{bmod {4}}end{aligned}}} Then {displaystyle left({frac {a}{b}}right)left({frac {b}{a}}right)=left({frac {a'}{b'}}right)left({frac {b'}{a'}}right).} Hilbert symbol The quadratic reciprocity law can be formulated in terms of the Hilbert symbol {displaystyle (a,b)_{v}} where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the Archimedean one and the p-adic absolute values for primes p). The Hilbert symbol {displaystyle (a,b)_{v}} is 1 or −1. It is defined to be 1 if and only if the equation {displaystyle ax^{2}+by^{2}=z^{2}} has a solution in the completion of the rationals at v other than {displaystyle x=y=z=0} . The Hilbert reciprocity law states that {displaystyle (a,b)_{v}} , for fixed a and b and varying v, is 1 for all but finitely many v and the product of {displaystyle (a,b)_{v}} over all v is 1. (This formally resembles the residue theorem from complex analysis.) The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol. There is no kind of reciprocity in the Hilbert reciprocity law; its name simply indicates the historical source of the result in quadratic reciprocity. Unlike quadratic reciprocity, which requires sign conditions (namely positivity of the primes involved) and a special treatment of the prime 2, the Hilbert reciprocity law treats all absolute values of the rationals on an equal footing. Therefore, it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extension can rightly be considered a generalization of quadratic reciprocity to all global fields.

Connection with cyclotomic fields The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.

Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class field theory. He wrote:[26] I confess that, as a student unaware of the history of the subject and unaware of the connection with cyclotomy, I did not find the law or its so-called elementary proofs appealing. I suppose, although I would not have (and could not have) expressed myself in this way that I saw it as little more than a mathematical curiosity, fit more for amateurs than for the attention of the serious mathematician that I then hoped to become. It was only in Hermann Weyl's book on the algebraic theory of numbers[27] that I appreciated it as anything more. Other rings There are also quadratic reciprocity laws in rings other than the integers.

Gaussian integers In his second monograph on quartic reciprocity[28] Gauss stated quadratic reciprocity for the ring {displaystyle mathbb {Z} [i]} of Gaussian integers, saying that it is a corollary of the biquadratic law in {displaystyle mathbb {Z} [i],} but did not provide a proof of either theorem. Dirichlet[29] showed that the law in {displaystyle mathbb {Z} [i]} can be deduced from the law for {displaystyle mathbb {Z} } without using quartic reciprocity.

For an odd Gaussian prime {displaystyle pi } and a Gaussian integer {displaystyle alpha } relatively prime to {displaystyle pi ,} define the quadratic character for {displaystyle mathbb {Z} [i]} by: {displaystyle left[{frac {alpha }{pi }}right]_{2}equiv alpha ^{frac {mathrm {N} pi -1}{2}}{bmod {pi }}={begin{cases}1&exists eta in mathbb {Z} [i]:alpha equiv eta ^{2}{bmod {pi }}\-1&{text{otherwise}}end{cases}}} Let {displaystyle lambda =a+bi,mu =c+di} be distinct Gaussian primes where a and c are odd and b and d are even. Then[30] {displaystyle left[{frac {lambda }{mu }}right]_{2}=left[{frac {mu }{lambda }}right]_{2},qquad left[{frac {i}{lambda }}right]_{2}=(-1)^{frac {b}{2}},qquad left[{frac {1+i}{lambda }}right]_{2}=left({frac {2}{a+b}}right).} Eisenstein integers Consider the following third root of unity: {displaystyle omega ={frac {-1+{sqrt {-3}}}{2}}=e^{frac {2pi imath }{3}}.} The ring of Eisenstein integers is {displaystyle mathbb {Z} [omega ].} [31] For an Eisenstein prime {displaystyle pi ,mathrm {N} pi neq 3,} and an Eisenstein integer {displaystyle alpha } with {displaystyle gcd(alpha ,pi )=1,} define the quadratic character for {displaystyle mathbb {Z} [omega ]} by the formula {displaystyle left[{frac {alpha }{pi }}right]_{2}equiv alpha ^{frac {mathrm {N} pi -1}{2}}{bmod {pi }}={begin{cases}1&exists eta in mathbb {Z} [omega ]:alpha equiv eta ^{2}{bmod {pi }}\-1&{text{otherwise}}end{cases}}} Let λ = a + bω and μ = c + dω be distinct Eisenstein primes where a and c are not divisible by 3 and b and d are divisible by 3. Eisenstein proved[32] {displaystyle left[{frac {lambda }{mu }}right]_{2}left[{frac {mu }{lambda }}right]_{2}=(-1)^{{frac {mathrm {N} lambda -1}{2}}{frac {mathrm {N} mu -1}{2}}},qquad left[{frac {1-omega }{lambda }}right]_{2}=left({frac {a}{3}}right),qquad left[{frac {2}{lambda }}right]_{2}=left({frac {2}{mathrm {N} lambda }}right).} Imaginary quadratic fields The above laws are special cases of more general laws that hold for the ring of integers in any imaginary quadratic number field. Let k be an imaginary quadratic number field with ring of integers {displaystyle {mathcal {O}}_{k}.} For a prime ideal {displaystyle {mathfrak {p}}subset {mathcal {O}}_{k}} with odd norm {displaystyle mathrm {N} {mathfrak {p}}} and {displaystyle alpha in {mathcal {O}}_{k},} define the quadratic character for {displaystyle {mathcal {O}}_{k}} as {displaystyle left[{frac {alpha }{mathfrak {p}}}right]_{2}equiv alpha ^{frac {mathrm {N} {mathfrak {p}}-1}{2}}{bmod {mathfrak {p}}}={begin{cases}1&alpha not in {mathfrak {p}}{text{ and }}exists eta in {mathcal {O}}_{k}{text{ such that }}alpha -eta ^{2}in {mathfrak {p}}\-1&alpha not in {mathfrak {p}}{text{ and there is no such }}eta \0&alpha in {mathfrak {p}}end{cases}}} for an arbitrary ideal {displaystyle {mathfrak {a}}subset {mathcal {O}}_{k}} factored into prime ideals {displaystyle {mathfrak {a}}={mathfrak {p}}_{1}cdots {mathfrak {p}}_{n}} define {displaystyle left[{frac {alpha }{mathfrak {a}}}right]_{2}=left[{frac {alpha }{{mathfrak {p}}_{1}}}right]_{2}cdots left[{frac {alpha }{{mathfrak {p}}_{n}}}right]_{2},} and for {displaystyle beta in {mathcal {O}}_{k}} define {displaystyle left[{frac {alpha }{beta }}right]_{2}=left[{frac {alpha }{beta {mathcal {O}}_{k}}}right]_{2}.} Let {displaystyle {mathcal {O}}_{k}=mathbb {Z} omega _{1}oplus mathbb {Z} omega _{2},} i.e. {displaystyle left{omega _{1},omega _{2}right}} is an integral basis for {displaystyle {mathcal {O}}_{k}.} For {displaystyle nu in {mathcal {O}}_{k}} with odd norm {displaystyle mathrm {N} nu ,} define (ordinary) integers a, b, c, d by the equations, {displaystyle {begin{aligned}nu omega _{1}&=aomega _{1}+bomega _{2}\nu omega _{2}&=comega _{1}+domega _{2}end{aligned}}} and a function {displaystyle chi (nu ):=imath ^{(b^{2}-a+2)c+(a^{2}-b+2)d+ad}.} If m = Nμ and n = Nν are both odd, Herglotz proved[33] {displaystyle left[{frac {mu }{nu }}right]_{2}left[{frac {nu }{mu }}right]_{2}=(-1)^{{frac {m-1}{2}}{frac {n-1}{2}}}chi (mu )^{m{frac {n-1}{2}}}chi (nu )^{-n{frac {m-1}{2}}}.} Also, if {displaystyle mu equiv mu '{bmod {4}}quad {text{and}}quad nu equiv nu '{bmod {4}}} Then[34] {displaystyle left[{frac {mu }{nu }}right]_{2}left[{frac {nu }{mu }}right]_{2}=left[{frac {mu '}{nu '}}right]_{2}left[{frac {nu '}{mu '}}right]_{2}.} Polynomials over a finite field Let F be a finite field with q = pn elements, where p is an odd prime number and n is positive, and let F[x] be the ring of polynomials in one variable with coefficients in F. If {displaystyle f,gin F[x]} and f is irreducible, monic, and has positive degree, define the quadratic character for F[x] in the usual manner: {displaystyle left({frac {g}{f}}right)={begin{cases}1&gcd(f,g)=1{text{ and }}exists h,kin F[x]{text{ such that }}g-h^{2}=kf\-1&gcd(f,g)=1{text{ and }}g{text{ is not a square}}{bmod {f}}\0&gcd(f,g)neq 1end{cases}}} If {displaystyle f=f_{1}cdots f_{n}} is a product of monic irreducibles let {displaystyle left({frac {g}{f}}right)=left({frac {g}{f_{1}}}right)cdots left({frac {g}{f_{n}}}right).} Dedekind proved that if {displaystyle f,gin F[x]} are monic and have positive degrees,[35] {displaystyle left({frac {g}{f}}right)left({frac {f}{g}}right)=(-1)^{{frac {q-1}{2}}(deg f)(deg g)}.} Higher powers Further information: Cubic reciprocity, Quartic reciprocity, Octic reciprocity, and Eisenstein reciprocity The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, Carl Gustav Jakob Jacobi, Gotthold Eisenstein, Richard Dedekind, Ernst Kummer, and David Hilbert to the study of general algebraic number fields and their rings of integers;[36] specifically Kummer invented ideals in order to state and prove higher reciprocity laws.

The ninth in the list of 23 unsolved problems which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the "Proof of the most general reciprocity law [f]or an arbitrary number field".[37] Building upon work by Philipp Furtwängler, Teiji Takagi, Helmut Hasse and others, Emil Artin discovered Artin reciprocity in 1923, a general theorem for which all known reciprocity laws are special cases, and proved it in 1927.[38] See also Dedekind zeta function Rational reciprocity law Zolotarev's lemma Notes ^ Gauss, DA § 4, arts 107–150 ^ E.g. in his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity). See facsimile page from Felix Klein's Development of Mathematics in the 19th century ^ See F. Lemmermeyer's chronology and bibliography of proofs in the external references ^ Veklych, Bogdan (2019). "A Minimalist Proof of the Law of Quadratic Reciprocity". The American Mathematical Monthly. 126 (10): 928. arXiv:2106.08121. doi:10.1080/00029890.2019.1655331. ^ Lemmermeyer, pp. 2–3 ^ Gauss, DA, art. 182 ^ Lemmermeyer, p. 3 ^ Lemmermeyer, p. 5, Ireland & Rosen, pp. 54, 61 ^ Ireland & Rosen, pp. 69–70. His proof is based on what are now called Gauss sums. ^ This section is based on Lemmermeyer, pp. 6–8 ^ The equivalence is Euler's criterion ^ The analogue of Legendre's original definition is used for higher-power residue symbols ^ E.g. Kronecker's proof (Lemmermeyer, ex. p. 31, 1.34) is to use Gauss's lemma to establish that {displaystyle left({frac {p}{q}}right)=operatorname {sgn} prod _{i=1}^{frac {q-1}{2}}prod _{k=1}^{frac {p-1}{2}}left({frac {k}{p}}-{frac {i}{q}}right)} and then switch p and q. ^ Gauss, DA, arts 108–116 ^ Gauss, DA, arts 117–123 ^ Gauss, DA, arts 130 ^ Gauss, DA, Art 131 ^ Gauss, DA, arts. 125–129 ^ Because the basic Gauss sum equals {displaystyle {sqrt {q^{*}}}.} ^ Because the quadratic field {displaystyle mathbb {Q} ({sqrt {q^{*}}})} is a subfield of the cyclotomic field {displaystyle mathbb {Q} (e^{frac {2pi i}{q}})} ^ Ireland & Rosen, pp 60–61. ^ Gauss, "Summierung gewisser Reihen von besonderer Art", reprinted in Untersuchumgen uber hohere Arithmetik, pp.463–495 ^ Lemmermeyer, Th. 2.28, pp 63–65 ^ Lemmermeyer, ex. 1.9, p. 28 ^ By Peter Gustav Lejeune Dirichlet in 1837 ^ "Archived copy" (PDF). Archived from the original (PDF) on January 22, 2012. Retrieved June 27, 2013. ^ Weyl, Hermann (1998). Algebraic Theory of Numbers. ISBN 0691059179. ^ Gauss, BQ § 60 ^ Dirichlet's proof is in Lemmermeyer, Prop. 5.1 p.154, and Ireland & Rosen, ex. 26 p. 64 ^ Lemmermeyer, Prop. 5.1, p. 154 ^ See the articles on Eisenstein integer and cubic reciprocity for definitions and notations. ^ Lemmermeyer, Thm. 7.10, p. 217 ^ Lemmermeyer, Thm 8.15, p.256 ff ^ Lemmermeyer Thm. 8.18, p. 260 ^ Bach & Shallit, Thm. 6.7.1 ^ Lemmermeyer, p. 15, and Edwards, pp.79–80 both make strong cases that the study of higher reciprocity was much more important as a motivation than Fermat's Last Theorem was ^ Lemmermeyer, p. viii ^ Lemmermeyer, p. ix ff References The Disquisitiones Arithmeticae has been translated (from Latin) into English and German. The German edition includes all of Gauss's papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n".

Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9 {{citation}}: |first2= has generic name (help) Gauss, Carl Friedrich; Maser, Hermann (translator into German) (1965), Untersuchungen über höhere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 {{citation}}: |first2= has generic name (help) 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 Untersuchungen über höhere Arithmetik.

Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Franz Lemmermeyer's Reciprocity Laws: From Euler to Eisenstein has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature citations for 196 different published proofs for the quadratic reciprocity law.

Kenneth Ireland and Michael Rosen's A Classical Introduction to Modern Number Theory also has many proofs of quadratic reciprocity (and many exercises), and covers the cubic and biquadratic cases as well. Exercise 13.26 (p. 202) says it all Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one.

Bach, Eric; Shallit, Jeffrey (1966), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02405-5 Edwards, Harold (1977), Fermat's Last Theorem, New York: Springer, ISBN 0-387-90230-9 Lemmermeyer, Franz (2000), Reciprocity Laws, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12893-0, ISBN 3-540-66957-4, MR 1761696 Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (second edition), New York: Springer, ISBN 0-387-97329-X External links "Quadratic reciprocity law", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Quadratic Reciprocity Theorem from MathWorld A play comparing two proofs of the quadratic reciprocity law A proof of this theorem at PlanetMath A different proof at MathPages F. Lemmermeyer's chronology and bibliography of proofs of the Quadratic Reciprocity Law (332 proofs) Categories: Algebraic number theoryModular arithmeticNumber theoryQuadratic residueTheorems in number theory

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