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020 _a9783540449492
_9978-3-540-44949-2
024 7 _a10.1007/b80624
_2doi
050 4 _aQA241-247.5
072 7 _aPBH
_2bicssc
072 7 _aMAT022000
_2bisacsh
072 7 _aPBH
_2thema
082 0 4 _a512.7
_223
245 1 4 _aThe Decomposition of Primes in Torsion Point Fields
_h[electronic resource] /
_cedited by Clemens Adelmann.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg,
_c2001.
300 _aVIII, 148 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v1761
505 0 _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.
520 _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.
650 0 _aNumber theory.
650 0 _aGeometry, algebraic.
650 1 4 _aNumber Theory.
_0http://scigraph.springernature.com/things/product-market-codes/M25001
650 2 4 _aAlgebraic Geometry.
_0http://scigraph.springernature.com/things/product-market-codes/M11019
700 1 _aAdelmann, Clemens.
_eeditor.
_4edt
_4http://id.loc.gov/vocabulary/relators/edt
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783662185209
776 0 8 _iPrinted edition:
_z9783540420354
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v1761
856 4 0 _uhttps://doi.org/10.1007/b80624
912 _aZDB-2-SMA
912 _aZDB-2-LNM
912 _aZDB-2-BAE
999 _c11441
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