03709nam 22006015 450 991014459880332120200702115509.03-540-44949-310.1007/b80624(CKB)1000000000233199(SSID)ssj0000322480(PQKBManifestationID)11268117(PQKBTitleCode)TC0000322480(PQKBWorkID)10289020(PQKB)11523247(DE-He213)978-3-540-44949-2(MiAaPQ)EBC3072990(PPN)155230077(EXLCZ)99100000000023319920121227d2001 u| 0engurnn|008mamaatxtccrThe Decomposition of Primes in Torsion Point Fields /by Clemens Adelmann1st ed. 2001.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,2001.1 online resource (VIII, 148 p.) Lecture Notes in Mathematics,0075-8434 ;1761Bibliographic Level Mode of Issuance: Monograph3-540-42035-5 Includes bibliographical references and index.Introduction -- Decomposition laws -- Elliptic curves -- Elliptic modular curves -- Torsion point fields -- Invariants and resolvent polynomials -- Appendix: Invariants of elliptic modular curves; L-series coefficients a p; Fully decomposed prime numbers; Resolvent polynomials; Free resolution of the invariant algebra.It is an historical goal of algebraic number theory to relate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively described in terms of the base ?eld. Suitable structures are the prime ideals of the ring of integers of the considered number ?eld. By examining the behaviouroftheprimeidealswhenembeddedintheextension?eld,su?cient information should be collected to distinguish the given extension from all other possible extension ?elds. The ring of integers O of an algebraic number ?eld k is a Dedekind ring. k Any non-zero ideal in O possesses therefore a decomposition into a product k of prime ideals in O which is unique up to permutations of the factors. This k decomposition generalizes the prime factor decomposition of numbers in Z Z. In order to keep the uniqueness of the factors, view has to be changed from elements of O to ideals of O . k k Given an extension K/k of algebraic number ?elds and a prime ideal p of O , the decomposition law of K/k describes the product decomposition of k the ideal generated by p in O and names its characteristic quantities, i. e. K the number of di?erent prime ideal factors, their respective inertial degrees, and their respective rami?cation indices. Whenlookingatdecompositionlaws,weshouldinitiallyrestrictourselves to Galois extensions. This special case already o?ers quite a few di?culties.Lecture Notes in Mathematics,0075-8434 ;1761Number theoryAlgebraic geometryNumber Theoryhttps://scigraph.springernature.com/ontologies/product-market-codes/M25001Algebraic Geometryhttps://scigraph.springernature.com/ontologies/product-market-codes/M11019Number theory.Algebraic geometry.Number Theory.Algebraic Geometry.512/.4Adelmann Clemensauthttp://id.loc.gov/vocabulary/relators/aut66298MiAaPQMiAaPQMiAaPQBOOK9910144598803321Decomposition of primes in torsion point fields377809UNINA