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Hilbert–Speiser Theorem
In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of , which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields. :Hilbert–Speiser Theorem. A finite abelian extension has a normal integral basis if and only if it is tamely ramified over . This is the condition that it should be a subfield of where is a squarefree odd number. This result was introduced by in his Zahlbericht and by . In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods. For example if we take a prime number , has a normal integral basis consisting of all the -th roots of unity other than . For a field contained in it, the field trace can be used to construct such a basis in also (see the article on Gaussian periods). Then in the case of squarefree and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of root of unity, roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods. History As the name suggests, the periods were introduced by Carl Friedrich Gauss, Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which : 2 \cos \left(\frac\right) = \zeta + \zeta^ \, is an example involving the seventeenth root of unity : \zeta = \exp \left(\frac\right). General definition Given an integer ''n'' > 1, let ''H'' be any subgroup of the multiplicative group : G = (\mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, Nigel Hitchin, and Thomas Schick. Currently, the managing editor of Mathematische Annalen is Yoshikazu Giga (University of Tokyo). Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947, the journal briefly ceased publication. References External links''Mathematische Annalen''homepage a ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invent Math
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compositum
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield. The tensor product of two fields is sometimes a field, and often a direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. Compositum of fields First, one defines the notion of the compositum of fields. This construction occurs frequently in field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must first specify a tower of fields. Let ''k'' be a field and ''L'' and ''K'' be two extensions of ''k''. The compositum, denoted ''K.L'', is defined to be K.L = k(K \cup L) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Trace
In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''. Definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. ''L'' can be viewed as a vector space over ''K''. Multiplication by ''α'', an element of ''L'', :m_\alpha:L\to L \text m_\alpha (x) = \alpha x, is a ''K''- linear transformation of this vector space into itself. The ''trace'', Tr''L''/''K''(''α''), is defined as the trace (in the linear algebra sense) of this linear transformation. For ''α'' in ''L'', let ''σ''(''α''), ..., ''σ''(''α'') be the roots (counted with multiplicity) of the minimal polynomial of ''α'' over ''K'' (in some extension field of ''K''). Then :\operatorname_(\alpha) = :K(\alpha)sum_^n\sigma_j(\alpha). If ''L''/''K'' is separable then each root appears only once (however this does not mean the coefficient above is one; for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roots Of Unity
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation z^n = 1. Unless otherwise specified, the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zahlbericht
In mathematics, the ''Zahlbericht'' (number report) was a report on algebraic number theory by . History In 1893 the German Mathematical Society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski would cover the more elementary parts of number theory while Hilbert would cover algebraic number theory. Minkowski eventually abandoned his report, while Hilbert's report was published in 1897. It was reprinted in volume 1 of his collected works, and republished in an English translation in 1998. and and the English introduction to give detailed discussions of the history and influence of Hilbert's ''Zahlbericht''. Some earlier reports on number theory include the report by H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in , and the report by . wrote an update of Hilbert's ''Zahlbericht'' that covered class field theory (republished in 1 volume as ). Contents Part 1 covers the theory of general number fields, including ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n)—and more precisely, because of the failure of unique factorization in their rings of integers—that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n \geq 1, let :\zeta_n=e^\in\C. This is a primitive nth root of unity. Then the nth cyclotomic field is the field extension \mathbb(\zeta_n) of \mathbb generated by \zeta_n. Properties * The nth cyclotomic polynomial :: \Phi_n(x) = \prod_\stackrel\!\!\! \left(x-e^\right) = \prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of \zeta_n o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
