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.

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.

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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