# Gauss’ Theorema Aureum: The Law of Quadratic Reciprocity

• Steve Wright
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2171)

## Abstract

Proposition  of Chap.  shows that the solution of the general second-degree congruence ax2 + bx + c ≡ 0 mod p for an odd prime p can be reduced to the solution of the congruence x2 ≡ b2 − 4ac mod p, and we also saw how the solution of x2 ≡ n mod m for a composite modulus m can be reduced by way of Gauss’ algorithm to the solution of x2 ≡ q mod p for prime numbers p and q. In this chapter, we will discuss a remarkable theorem known as the Law of Quadratic Reciprocity, which provides a very powerful method for determining the solvability of congruences of this last type. The theorem states that if p and q are distinct odd primes then the congruences x2 ≡ q mod p and x2 ≡ p mod q are either both solvable or both not solvable, unless p and q are both congruent to 3 mod 4, in which case one is solvable and the other is not. As a simple but no less striking example of the power of this theorem, suppose one wants to know if x2 ≡ 5 mod 103 has any solutions. Since 5 is not congruent to 3 mod 4, the quadratic reciprocity law asserts that x2 ≡ 5 mod 103 and x2 ≡ 103 mod 5 are both solvable or both not. But solution of the latter congruence reduces to x2 ≡ 3 mod 5, which clearly has no solutions. Hence neither does x2 ≡ 5 mod 103.

## Keywords

Galois Group Algebraic Number Nonzero Ideal Algebraic Integer Quadratic Residue
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Bibliography

1. 1.
B. Berndt, Classical theorems on quadratic residues. Enseignement Math. 22, 261–304 (1976)
2. 2.
H. Cohen, Number Theory, vol. I (Springer, New York, 2000)
3. 3.
J.B. Conway, Functions of One Complex Variable, vol. 1 (Springer, New York, 1978)
4. 4.
K.L. Chung, A Course in Probability Theory (Academic Press, New York, 1974)
5. 5.
H. Davenport, On character sums in finite fields. Acta Math. 71, 99–121 (1939)
6. 6.
H. Davenport, Multiplicative Number Theory (Springer, New York, 2000)
7. 7.
H. Davenport, P. Erdös, The distribution of quadratic and higher residues. Publ. Math. Debrecen 2, 252–265 (1952)
8. 8.
R. Dedekind, Sur la Th $$\acute{\text{e}}$$ orie des Nombres Entiers Alg $$\acute{\text{e}}$$ briques (1877); English translation by J. Stillwell (Cambridge University Press, Cambridge, 1996)Google Scholar
9. 9.
P.G.L. Dirichlet, Sur la convergence des series trigonom$$\acute{\text{e}}$$ trique qui servent $$\grave{\text{a}}$$ repr$$\acute{\text{e}}$$ senter une fonction arbitraire entre des limites donn$$\acute{\text{e}}$$ e. J. Reine Angew. Math. 4, 157–169 (1829)Google Scholar
10. 10.
P.G.L. Dirichlet, Beweis eines Satzes da$$\ss$$ jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enh$$\ddot{\text{a}}$$ lt. Abh. K. Preuss. Akad. Wiss. 45–81 (1837)Google Scholar
11. 11.
P.G.L. Dirichlet, Recherches sur diverses applications de l’analyse infinit$$\acute{\text{e}}$$ simal $$\grave{\text{a}}$$ la th$$\acute{\text{e}}$$ orie des nombres. J. Reine Angew. Math. 19, 324–369 (1839); 21 (1–12), 134–155 (1840)Google Scholar
12. 12.
P.G.L. Dirichlet, Vorlesungen über Zahlentheorie (1863); English translation by J. Stillwell (American Mathematical Society, Providence, 1991)Google Scholar
13. 13.
J. Dugundji, Topology (Allyn and Bacon, Boston, 1966)
14. 14.
P. Erdös, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A. 35, 374–384 (1949)
15. 15.
L. Euler, Theoremata circa divisores numerorum in hac forma pa 2 ± qb 2 contentorum. Comm. Acad. Sci. Petersburg 14, 151–181 (1744/1746)Google Scholar
16. 16.
L. Euler, Theoremata circa residua ex divisione postestatum relicta. Novi Commet. Acad. Sci. Petropolitanea 7, 49–82 (1761)Google Scholar
17. 17.
L. Euler, Observationes circa divisionem quadratorum per numeros primes. Opera Omnia I-3, 477–512 (1783)Google Scholar
18. 18.
M. Filaseta, D. Richman, Sets which contain a quadratic residue modulo p for almost all p. Math. J. Okayama Univ. 39, 1–8 (1989)Google Scholar
19. 19.
C.F. Gauss, Disquisitiones Arithmeticae (1801); English translation by A. A. Clarke (Springer, New York, 1986)Google Scholar
20. 20.
C.F. Gauss, Theorematis arithmetici demonstratio nova. Göttingen Comment. Soc. Regiae Sci. XVI, 8 pp. (1808)Google Scholar
21. 21.
C.F. Gauss, Summatio serierum quarundam singularium. Göttingen Comment. Soc. Regiae Sci. 36 pp. (1811)Google Scholar
22. 22.
C.F. Gauss, Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et amplicationes novae, 1818, Werke, vol. II (Georg Olms Verlag, Hildescheim, 1973), pp. 47–64Google Scholar
23. 23.
C.F. Gauss, Theorematis fundamentallis in doctrina residuis demonstrationes et amplicationes novae. Göttingen Comment. Soc. Regiae Sci. 4, 17 pp. (1818)Google Scholar
24. 24.
C.F. Gauss, Theoria residuorum biquadraticorum: comentatio prima. Göttingen Comment. Soc. Regiae Sci. 6, 28 pp. (1828)Google Scholar
25. 25.
C.F. Gauss, Theoria residuorum biquadraticorum: comentatio secunda. Göttingen Comment. Soc. Regiae Sci. 7, 56 pp. (1832)Google Scholar
26. 26.
D. Gröger, Gauß’ Reziprozitätgesetze der Zahlentheorie: Eine Gesamtdarstellung der Hinterlassenschaft in Zeitgemäßer Form (Erwin-Rauner Verlag, Augsburg, 2013)Google Scholar
27. 27.
E. Hecke, Vorlesungen über die Theorie der Algebraischen Zahlen (1923); English translation by G. Brauer and J. Goldman (Springer, New York, 1981)
28. 28.
D. Hilbert, Die Theorie der Algebraischen Zahlkörper (1897); English translation by I. Adamson (Springer, Berlin, 1998)Google Scholar
29. 29.
T. Hungerford, Algebra (Springer, New York, 1974)
30. 30.
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory (Springer, New York, 1990)
31. 31.
P. Kurlberg, The distribution of spacings between quadratic residues II. Isr. J. Math. 120, 205–224 (2000)
32. 32.
P. Kurlberg, Z. Rudnick, The distribution of spacings between quadratic residues. Duke Math. J. 100, 211–242 (1999)
33. 33.
J.L. Lagrange, Probl$$\grave{\text{e}}$$ mes ind$$\acute{\text{e}}$$ termin$$\acute{\text{e}}$$ s du second degr$$\acute{\text{e}}$$. Mém. Acad. R. Berlin 23, 377–535 (1769)Google Scholar
34. 34.
J.L. Lagrange, Reserches d’Arithm$$\acute{\text{e}}$$ tique, 2nde partie. Nouv. Mém. Acad. Berlin 349–352 (1775)Google Scholar
35. 35.
E. Landau, Elementary Number Theory. English translation by J. Goodman (Chelsea, New York, 1958)
36. 36.
A. Legendre, Reserches d’analyse indéterminée. Histoiré de l’Académie Royale des Sciences de Paris (1785), pp. 465–559, Paris 1788Google Scholar
37. 37.
A. Legendre, Essai sur la Th $$\acute{\text{e}}$$ orie des Nombres (Paris, 1798)Google Scholar
38. 38.
F. Lemmermeyer, Reciprocity Laws (Springer, New York/Berlin/Heidelberg, 2000)
39. 39.
40. 40.
D. Marcus, Number Fields (Springer, New York, 1977)
41. 41.
H. Montgomery, R. Vaughan, Multiplicative Number Theory I: Classical Theory (Cambridge University Press, Cambridge, 2007)
42. 42.
43. 43.
O. Ore, Les Corps Alg $$\acute{\text{e}}$$ briques et la Th $$\acute{\text{e}}$$ orie des Id $$\acute{\text{e}}$$ aux (Gauthier-Villars, Paris, 1934)Google Scholar
44. 44.
G. Perel’muter, On certain character sums. Usp. Mat. Nauk. 18, 145–149 (1963)
45. 45.
C. de la Vall$$\acute{\text{e}}$$ e Poussin, Recherches analytiques sur la th$$\acute{\text{e}}$$ orie des nombres premiers. Ann. Soc. Sci. Bruxelles 20, 281–362 (1896)Google Scholar
46. 46.
H. Rademacher, Lectures on Elementary Number Theory (Krieger, New York, 1977)
47. 47.
G.F.B. Riemann, $$\ddot{\text{U}}$$ ber die Anzahl der Primzahlen unter einer gegebenen Gr$$\ddot{\text{o}}\ss$$ e. Monatsberischte der Berlin Akademie (1859), pp. 671–680Google Scholar
48. 48.
K. Rosen, Elementary Number Theory and Its Applications (Pearson, Boston, 2005)Google Scholar
49. 49.
J. Rosenberg, Algebraic K-Theory and Its Application (Springer, New York, 1996)Google Scholar
50. 50.
W. Schmidt, Equations over Finite Fields: an Elementary Approach (Springer, Berlin, 1976)
51. 51.
A. Selberg, An elementary proof of Dirichlet’s theorem on primes in arithmetic progressions, Ann. Math. 50, 297–304 (1949)
52. 52.
A. Selberg, An elementary proof of the prime number theorem. Ann. Math. 50, 305–313 (1949)
53. 53.
A. Shamir, Identity-based cryptosystems and signature schemes, in Advances in Cryptology, ed. by G.R. Blakely, D. Chaum (Springer, Berlin, 1985), pp. 47–53
54. 54.
J. Shohat, J.D. Tamarkin, The Problem of Moments (American Mathematical Society, New York, 1943)
55. 55.
R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141, 553–572 (1995)
56. 56.
J. Urbanowicz, K.S. Williams, Congruences for L-Functions (Kluwer, Dordrecht, 2000)
57. 57.
A. Weil, Sur les Courbes Algébriques et les Variétes qui s’en Déduisent (Hermann et Cie, Paris, 1948)
58. 58.
A. Weil, Basic Number Theory (Springer, New York, 1973)
59. 59.
L. Weisner, Introduction to the Theory of Equations (MacMillan, New York, 1938)
60. 60.
A. Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. Math. 141, 443–551 (1995)
61. 61.
S. Wright, Quadratic non-residues and the combinatorics of sign multiplication. Ars Combin. 112, 257–278 (2013)
62. 62.
S. Wright, Quadratic residues and non-residues in arithmetic progression. J. Number Theory 133, 2398–2430 (2013)
63. 63.
S. Wright, On the density of primes with a set of quadratic residues or non-residues in given arithmetic progression, J. Combin. Number Theory 6, 85–111 (2015)
64. 64.
B.F. Wyman, What is a reciprocity law? Am. Math. Monthly 79, 571–586 (1972)
65. 65.
A. Zygmund, Trigonometric Series (Cambridge University Press, Cambridge, 1968)