• Thumbnail for Quadratic reciprocity
    theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime...
    111 KB (8,566 words) - 03:50, 12 March 2025
  • In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f ( x ) {\displaystyle...
    13 KB (1,830 words) - 13:11, 25 May 2025
  • In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred...
    22 KB (4,009 words) - 08:35, 9 May 2025
  • 1797 or 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet...
    43 KB (2,477 words) - 10:47, 29 May 2025
  • Thumbnail for Floor and ceiling functions
    introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth...
    37 KB (5,912 words) - 20:14, 22 April 2025
  • nonresidue. The first supplement to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then...
    54 KB (5,539 words) - 21:19, 19 January 2025
  • a quadratic residue or nonresidue modulo n {\displaystyle n} . The Kronecker symbol also satisfies the following versions of quadratic reciprocity law...
    13 KB (1,722 words) - 00:56, 18 November 2024
  • density theorem. The law of quadratic reciprocity implies that the splitting behaviour of a prime p {\displaystyle p} in a quadratic field depends only on p...
    12 KB (1,306 words) - 15:52, 8 June 2025
  • Thumbnail for Algebraic number theory
    {q-1}{2}}}.} A reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early...
    40 KB (5,798 words) - 10:21, 25 April 2025
  • Thumbnail for Number theory
    seems to have come from the study of higher reciprocity laws, that is, generalizations of quadratic reciprocity. Number fields are often studied as extensions...
    95 KB (12,176 words) - 01:29, 10 June 2025
  • Thumbnail for Carl Friedrich Gauss
    the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the...
    181 KB (17,930 words) - 03:08, 11 June 2025
  • and applied them to quadratic, cubic, and biquadratic reciprocity laws. For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is...
    8 KB (1,660 words) - 09:12, 17 October 2024
  • term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and...
    16 KB (2,326 words) - 06:37, 14 April 2025
  • ISBN 9780521585330 Russinoff, David M. (1992), "A Mechanical Proof of Quadratic Reciprocity", J. Autom. Reason., 8 (1): 3–21, doi:10.1007/BF00263446, S2CID 14824949...
    14 KB (1,399 words) - 16:29, 2 June 2025
  • Thumbnail for Prime number
    prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence...
    117 KB (14,179 words) - 21:25, 8 June 2025
  • Thumbnail for Fundamental theorem of arithmetic
    all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished...
    23 KB (3,274 words) - 10:44, 5 June 2025
  • residue r is a quadratic residue (mod q) if and only if it is a biquadratic residue (mod q). Indeed, the first supplement of quadratic reciprocity states that...
    30 KB (4,817 words) - 08:05, 9 May 2024
  • Modular_arithmetic#Residue_class Quadratic_residue#Prime_power_modulus Artin symbol Gauss's lemma Quadratic reciprocity deals with squares; higher refers...
    11 KB (1,321 words) - 06:01, 8 December 2023
  • point of the program was Emil Artin's reciprocity law, which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of...
    21 KB (2,351 words) - 22:52, 31 May 2025
  • characteristics Helmholtz reciprocity, linear propagation. Reciprocity law (law of reciprocity) in mathematics, including Quadratic reciprocity, a fundamental result...
    5 KB (635 words) - 21:54, 20 January 2025
  • Carl Friedrich Gauss referred to the law of quadratic reciprocity as the "fundamental theorem" of quadratic residues. There are also a number of "fundamental...
    5 KB (553 words) - 13:53, 14 September 2024
  • in time O((log n)²) using Jacobi's generalization of the law of quadratic reciprocity. Given an odd number n one can contemplate whether or not the congruence...
    10 KB (1,517 words) - 18:46, 16 April 2025
  • 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...
    26 KB (4,061 words) - 14:25, 26 March 2024
  • number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it...
    22 KB (2,911 words) - 15:09, 24 April 2025
  • Reciprocity theorem may refer to: Quadratic reciprocity, a theorem about modular arithmetic Cubic reciprocity Quartic reciprocity Artin reciprocity Weil...
    913 bytes (124 words) - 07:44, 2 March 2023
  • Gaussian integer, Gaussian rational Quadratic field Cyclotomic field Cubic field Biquadratic field Quadratic reciprocity Ideal class group Dirichlet's unit...
    2 KB (187 words) - 23:15, 29 June 2024
  • and martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern...
    3 KB (431 words) - 23:54, 14 December 2024
  • theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher...
    9 KB (1,578 words) - 01:51, 24 April 2025
  • {\displaystyle -1} for the remaining. It is easy to compute using the law of quadratic reciprocity in a manner akin to the Euclidean algorithm; see Legendre symbol...
    7 KB (1,204 words) - 20:32, 20 December 2023
  • Gauss's lemma (number theory) (category Quadratic residue)
    quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity...
    18 KB (3,199 words) - 10:28, 5 November 2024