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Thaine's Theorem
In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by . Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem , to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem . Formulation Let p and q be distinct odd primes with q not dividing p-1. Let G^+ be the Galois group of F=\mathbb Q(\zeta_p^+) over \mathbb, let E be its group of units, let C be the subgroup of cyclotomic units, and let Cl^+ be its class group. If \theta\in\mathbb Z ^+/math> annihilates E/CE^q then it annihilates Cl^+/Cl^. References * See in particular Chapter 14 (pp. 91–94) for the use of Thaine's theorem to prove Mihăilescu's theorem Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stickelberger's Theorem
In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890). The Stickelberger element and the Stickelberger ideal Let denote the th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the th roots of unity to \mathbb (where is an integer). It is a Galois extension of \mathbb with Galois group isomorphic to the multiplicative group of integers modulo . The Stickelberger element (of level or of ) is an element in the group ring and the Stickelberger ideal (of level or of ) is an ideal in the group ring . They are defined as follows. Let denote a primitive th root of unity. The isomorphism from to is given by sending to defined by the relation :\sigma_a(\zeta_m) = \zeta_m^a. The Stickelberger element of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mazur–Wiles Theorem
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between ''p''-adic ''L''-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by . The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields,, CM fields, elliptic curves, and so on. Motivation was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian variety. In this analogy, * The action of the Frobenius corresponds to the action of the group Γ. * The Jacobian of a curve corresponds to a module ''X'' over Γ defined in terms of ideal class groups. * The zeta function of a curve over a finite field corresponds to a ''p''-adic ''L''-func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate–Shafarevich Group
In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of (i.e. the -adic fields obtained from , as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as :\bigcap_v\mathrm\left(H^1\left(G_K,A\right)\rightarrow H^1\left(G_,A_v\right)\right). This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation , where is the Cyrillic letter " Sha", for Shafarevich, replacing the older notation or . Elements of the Tate–Shafarevich group Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of that have -rational points for every place of , but no -rational point. Thus, the group measures the extent to which the Hasse principle fails to ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mihăilescu's Theorem
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the ''only'' case of two consecutive perfect powers. That is to say, that History The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (''x'', ''y'') was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case ''b'' = 2. In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding ''x'',''y'' in terms of ''a'', ''b'' to give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Fields
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , 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 ) – 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 , let ; this is a primitive th root of unity. Then the th cyclotomic field is the extension of generated by . Properties * The th cyclotomic polynomial : \Phi_n(x) = \!\!\!\prod_\stackrel\!\!\! \left(x-e^\right) = \!\!\!\prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of over . * The conjugates of in are therefore the other primiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |