The Decomposition of Primes in Torsion Point Fields

Clemens Adelmann (Autor)

Buch | Softcover

VIII, 148 Seiten

2001
Springer Berlin (Verlag)
978-3-540-42035-4 (ISBN)

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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.

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.

Erscheint lt. Verlag	22.5.2001
Reihe/Serie	Lecture Notes in Mathematics
Zusatzinfo	VIII, 148 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	239 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Algebra • algebraic number theory • Elliptic Curve • Invariant theory • modular form • Number Theory • Prime number • Torsion
ISBN-10	3-540-42035-5 / 3540420355
ISBN-13	978-3-540-42035-4 / 9783540420354
Zustand	Neuware