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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
In dem Buch wird die Theorie elliptischer Funktionen und ihrer
Anwendungen ausfuhrlich dargestellt. Beginnend mit numerischen
Berechnungen enthalt Abschnitt I geometrische, Abschnitt II
arithmetische Anwendungen. Wichtige Ziele sind ein Beweis des
Satzes von Abel und die Berechnung der Klasseninvarianten . Weitere
Anwendungen sind die Berechnung von Darstellungsanzahlen
quadratischer Formen und die Bestimmung von Klassenzahlrelationen.
Der (unvollendete) physikalische Teil des Werks widmet sich der
analytischen Theorie des ebenen Gelenkvierecks.
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