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Eisenstein Sum
In mathematics, an Eisenstein sum is a finite sum depending on a finite field and related to a Gauss sum. Eisenstein sums were introduced by Eisenstein in 1848, named "Eisenstein sums" by Stickelberger in 1890, and rediscovered by Yamamoto in 1985, who called them relative Gauss sums. Definition The Eisenstein sum is given by :E(\chi,\alpha)=\sum_\chi(t) where ''F'' is a finite extension of the finite field ''K'', and χ is a character of the multiplicative group of ''F'', and α is an element of ''K''. References Bibliography * * * * * *{{Citation , last1=Yamamoto , first1=K. , title=Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) , publisher=World Sci. Publishing , location=Singapore , mr=827799 , zbl=0634.12017 , year=1985 , chapter=On congruences arising from relative Gauss sums , pages=423–446 Algebraic number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Sum
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a group homomorphism of the additive group into the unit circle, and is a group homomorphism of the unit group into the unit circle, extended to non-unit , where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function. Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet -functions, where for a Dirichlet character the equation relating and ) (where is the complex conjugate of ) involves a factor :\frac. History The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for the field of residues modulo a prime number , and the Legendre symbol. In this case Gauss proved that or for congruent to 1 or 3 m ... [...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] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
