theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime...
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In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f ( x ) {\displaystyle...
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In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred...
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Legendre symbol (redirect from Quadratic residue symbol)
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
introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth...
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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
Kronecker symbol (section Quadratic reciprocity)
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...
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Algebraic number theory (section Reciprocity laws)
{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
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
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...
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term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and...
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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...
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prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence...
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all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished...
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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...
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Power residue symbol (redirect from Power reciprocity law)
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
Langlands program (redirect from Langlands reciprocity)
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...
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Carl Friedrich Gauss referred to the law of quadratic reciprocity as the "fundamental theorem" of quadratic residues. There are also a number of "fundamental...
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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...
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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...
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number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it...
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Reciprocity theorem may refer to: Quadratic reciprocity, a theorem about modular arithmetic Cubic reciprocity Quartic reciprocity Artin reciprocity Weil...
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Gaussian integer, Gaussian rational Quadratic field Cyclotomic field Cubic field Biquadratic field Quadratic reciprocity Ideal class group Dirichlet's unit...
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and martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern...
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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...
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{\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...
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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