LEADER 03734nam 22006015 450 001 996466380003316 005 20200702115509.0 010 $a3-540-44949-3 024 7 $a10.1007/b80624 035 $a(CKB)1000000000233199 035 $a(SSID)ssj0000322480 035 $a(PQKBManifestationID)11268117 035 $a(PQKBTitleCode)TC0000322480 035 $a(PQKBWorkID)10289020 035 $a(PQKB)11523247 035 $a(DE-He213)978-3-540-44949-2 035 $a(MiAaPQ)EBC3072990 035 $a(PPN)155230077 035 $a(EXLCZ)991000000000233199 100 $a20121227d2001 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 14$aThe Decomposition of Primes in Torsion Point Fields$b[electronic resource] /$fby Clemens Adelmann 205 $a1st ed. 2001. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2001. 215 $a1 online resource (VIII, 148 p.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v1761 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-540-42035-5 320 $aIncludes bibliographical references and index. 327 $aIntroduction -- 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. 330 $aIt is an historical goal of algebraic number theory to relate all algebraic extensionsofanumber?eldinauniquewaytostructuresthatareexclusively described in terms of the base ?eld. Suitable structures are the prime ideals of the ring of integers of the considered number ?eld. By examining the behaviouroftheprimeidealswhenembeddedintheextension?eld,su?cient information should be collected to distinguish the given extension from all other possible extension ?elds. The ring of integers O of an algebraic number ?eld k is a Dedekind ring. k Any non-zero ideal in O possesses therefore a decomposition into a product k of prime ideals in O which is unique up to permutations of the factors. This k decomposition generalizes the prime factor decomposition of numbers in Z Z. In order to keep the uniqueness of the factors, view has to be changed from elements of O to ideals of O . k k Given an extension K/k of algebraic number ?elds and a prime ideal p of O , the decomposition law of K/k describes the product decomposition of k the ideal generated by p in O and names its characteristic quantities, i. e. K the number of di?erent prime ideal factors, their respective inertial degrees, and their respective rami?cation indices. Whenlookingatdecompositionlaws,weshouldinitiallyrestrictourselves to Galois extensions. This special case already o?ers quite a few di?culties. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v1761 606 $aNumber theory 606 $aAlgebraic geometry 606 $aNumber Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M25001 606 $aAlgebraic Geometry$3https://scigraph.springernature.com/ontologies/product-market-codes/M11019 615 0$aNumber theory. 615 0$aAlgebraic geometry. 615 14$aNumber Theory. 615 24$aAlgebraic Geometry. 676 $a512/.4 700 $aAdelmann$b Clemens$4aut$4http://id.loc.gov/vocabulary/relators/aut$066298 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466380003316 996 $aDecomposition of primes in torsion point fields$9377809 997 $aUNISA