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Octic Reciprocity Law
In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity. There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol \left(\frac xp\right)_k to be +1 if ''x'' is a ''k''-th power modulo the prime ''p'' and -1 otherwise. Let ''p'' and ''q'' be distinct primes congruent to 1 modulo 8, such that \left(\frac pq\right)_4 = \left(\frac qp\right)_4 = +1 . Let ''p'' = ''a''2 + ''b''2 = ''c''2 + 2''d''2 and ''q'' = ''A''2 + ''B''2 = ''C''2 + 2''D''2, with ''aA'' odd. Then : \left(\frac pq\right)_8 \left(\frac qp\right)_8 = \left(\fracq\right)_4 \left(\fracq\right)_2 \ . See also * Artin reciprocity * Eisenstein reciprocity In algebraic number 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 powers. It is one o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocity Law
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f(x) with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial f(x) = x^2 + ax + b splits into linear terms when reduced mod p. That is, it determines for which prime numbers the relationf(x) \equiv f_p(x) = (x-n_p)(x-m_p) \text (\text p)holds. For a general reciprocity lawpg 3, it is defined as the rule determining which primes p the polynomial f_p splits into linear factors, denoted \text\. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (''p''/''q'') generalizing the quadratic reciprocity symbol, that describes when a prime number is an ''n''th power residue modulo another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). Hilbert ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Quadratic Reciprocity
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: 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 x^2\equiv a \bmod p for an odd prime p; that is, to determine the "perfect squares" modulo 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 p\equiv 3 \bmod 4 using Euler's criterion one can give an explicit formula for the "square roots" modulo p of a quadratic residue a, namely, :\pm a^ indeed, :\left (\pm a^ \right )^2=a^=a\cdot a^\equiv a\left(\frac\right)=a \bmod p. This formula only works if it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''3 ≡ ''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 ''x''3 ≡ ''p'' (mod ''q'') is solvable if and only if ''x''3 ≡ ''q'' (mod ''p'') is solvable. History Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, 62 years after his death. Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones 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 techniques ( Gauss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence ''x''4 ≡ ''p'' (mod ''q'') to that of ''x''4 ≡ ''q'' (mod ''p''). History Euler made the first conjectures about biquadratic reciprocity. Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He saidGauss, BQ, § 67 that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37. The first published proofs were by Ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Reciprocity Law
In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity. As an example, there are rational biquadratic and octic reciprocity law In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity. There is a rational reciprocity law for 8th pow ...s. Define the symbol (''x'', ''p'')''k'' to be +1 if ''x'' is a ''k''-th power modulo the prime ''p'' and -1 otherwise. Let ''p'' and ''q'' be distinct primes congruent to 1 modulo 4, such that (''p'', ''q'')2 = (''q'', ''p'')2 = +1. Let ''p'' = ''a''2 + ''b''2 and ''q'' = ''A''2 + ''B''2 with ''aA'' odd. Then : (p, q)_4 (q, p)_4 = (-1)^ (aB-bA, q)_2 \ . If in addition ''p'' and ''q'' are congruent to 1 modulo 8, let ''p'' = ''c''2 + 2''d''2 and ''q'' = ''C''2 + 2''D''2. Then : (p, q)_8 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Reciprocity
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term " reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem. Statement Let L/K be a Galois extension of global fields and C_L stand for the idèle class group of L. One of the statements of the Artin reciprocity law is that there is a canonical isomorphism called the global symbol mapNeukirch (1999) p.391 : \theta: C_K/ \to \operatorname(L/K)^, where \text denotes the abelianization of a group, and \operatorname(L/K) is the Galois group of L over K. The map \theta is defined by assembling the maps called the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Reciprocity
In algebraic number 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 powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by , though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839. Background and notation Let m > 1 be an integer, and let \mathcal_m be the ring of integers of the ''m''-th cyclotomic field \mathbb(\zeta_m), where \zeta_m=e^ is a primitive ''m''-th root of unity. The numbers \zeta_m, \zeta_m^2,\dots\zeta_m^m=1 are units in \mathcal_m. (There are other units as well.) Primary numbers A number \alpha\in\mathcal_m is called primary if it is not a unit, is relatively prime to m, and is congruent to a rational (i.e. in \mathbb) integer (\!\bmo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific Journal Of Mathematics
The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley. It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription. The journal is incorporated as a 501(c)(3) organization A 501(c)(3) organization is a United States corporation, Trust (business), trust, unincorporated association or other type of organization exempt from federal income tax under section 501(c)(3) of Title 26 of the United States Code. It is one of .... The 255-page proof of the odd order theorem, by Walter Feit and John Griggs Thompson, was publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
