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Haran's Diamond Theorem
In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian. Statement of the diamond theorem Let ''K'' be a Hilbertian field and ''L'' a separable extension of ''K''. Assume there exist two Galois extensions ''N'' and ''M'' of ''K'' such that ''L'' is contained in the compositum ''NM'', but is contained in neither ''N'' nor ''M''. Then ''L'' is Hilbertian. The name of the theorem comes from the pictured diagram of fields, and was coined by Jarden. Some corollaries Weissauer's theorem This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof led Haran to his diamond theorem. ;Weissauer's theorem Let ''K'' be a Hilbertian field, ''N'' a Galois extension of ''K'', and ''L'' a finite proper extension of ''N''. Then ''L'' is Hilbertian. ;Proof using the diamond theorem If ''L'' is finite over ''K'', it is ... [...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] |
Hilbertian Field
In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case; solving within ''K'' a polynomial that does not always factorise. One is also allowed to take finite unions. Formulation More precisely, let ''V'' be an algebraic variety over ''K'' (assumptions here are: ''V'' is an irreducible set, a quasi-projective variety, and ''K'' has characteristic zero). A type I thin set is a subset of ''V''(''K'') that is not Zariski-dense. That means it lies in an algebraic set that is a finite union of algebraic varieties of dimension lower than ''d'', the dimension of ''V''. A type II thin set is an image of an algebraic morphism (essentially a polynomial mapping) φ, applied to the ''K''-points of some other ''d''-dimensional al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond Theorem
In mathematics, diamond theorem may refer to: * Aztec diamond theorem on tilings * Diamond isomorphism theorem on modular lattices * Haran's diamond theorem on Hilbertian fields * Second Isomorphism Theorem for Groups * Cullinane diamond theorem on the Galois geometry Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a'' Ga ... of graphic patterns See also * Diamond (other) {{mathdab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Irreducibility Theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory. Formulation of the theorem Hilbert's irreducibility theorem. Let :f_1(X_1, \ldots, X_r, Y_1, \ldots, Y_s), \ldots, f_n(X_1, \ldots, X_r, Y_1, \ldots, Y_s) be irreducible polynomials in the ring :\Q(X_1, \ldots, X_r) _1, \ldots, Y_s Then there exists an ''r''-tuple of rational numbers (''a''1, ..., ''ar'') such that :f_1(a_1, \ldots, a_r, Y_1,\ldots, Y_s), \ldots, f_n(a_1, \ldots, a_r, Y_1,\ldots, Y_s) are irreducible in the ring :\Q _1,\ldots, Y_s Remarks. * It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible special ... [...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] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Algebra
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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