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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 In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth degre ... and octic reciprocity laws. 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)_ ... [...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 refo ... [...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 Eise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octic Reciprocity Law
In number theory, octic reciprocity is a 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 irr ... relating the residues of 8th powers modular arithmetic, modulo prime number, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–1928 Kurt Hens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...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. References Mathematics journals Publications established in 1951 Mathematical Sciences Publishers academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |