Conditions under which the congruence x^3 equals p (mod q) is solvable
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable.
Cubicreciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q)...
curve C defined by a cubic equation Cubicreciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocityCubic surface, an algebraic...
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f ( x ) {\displaystyle...
Look up reciprocity in Wiktionary, the free dictionary. Reciprocity may refer to: Reciprocity (Canadian politics), free trade with the United States of...
Reciprocity theorem may refer to: Quadratic reciprocity, a theorem about modular arithmetic Cubicreciprocity Quartic reciprocity Artin reciprocity Weil...
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations...
Similarly, cubicreciprocity relates the solvability of x3 ≡ q (mod p) to that of x3 ≡ p (mod q), and biquadratic (or quartic) reciprocity is a relation...
and multiplication by units). Similarly, in 1844 while working on cubicreciprocity, Eisenstein introduced the ring Z [ ω ] {\displaystyle \mathbb {Z}...
theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubicreciprocity law to residues of higher...
}\right)}}.} Gauss sums can be used to prove quadratic reciprocity, cubicreciprocity, and quartic reciprocity. Gauss sums can be used to calculate the number...
two proofs of the law of quadratic reciprocity, and the analogous laws of cubicreciprocity and quartic reciprocity. In June 1844 Eisenstein visited Carl...
reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubicreciprocity,...
symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws. Let k be an algebraic number field with...
(UFD) and proved the biquadratic reciprocity law. Jacobi and Eisenstein at around the same time proved a cubicreciprocity law for the Eisenstein integers...
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred...
instead of the sine function, Eisenstein was able to prove cubic and quartic reciprocity as well. The Jacobi symbol (a/n) is a generalization of the...
who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws. For an odd prime number p and an integer a, the quadratic...
equation of degree 4 Quartic curve, an algebraic curve of degree 4 Quartic reciprocity, a theorem from number theory Quartic surface, a surface defined by an...
Gaussian rational Quadratic field Cyclotomic field Cubic field Biquadratic field Quadratic reciprocity Ideal class group Dirichlet's unit theorem Discriminant...
residue is a 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...
useful for the analysis of Brownian motion and martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a...
Discrete q-Hermite polynomials Wiener–Hermite expansion Hermite reciprocity, a reciprocity law concerning covariants of binary forms Hermite ring, a ring...
seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity. Number fields are often studied as extensions...
law of biquadratic reciprocity – as Gauss discovered, rings of complex integers are the natural setting for such higher reciprocity laws. In the second...
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