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.

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.

Der mathematische Teil des Werkes beginnt mit numerischen Berechnungen im Gebiet der elliptischen Funktionen. Abschnitt I enthält einen bunten Strauß geometrischer Anwendungen, z.B. Lemniskatenteilung, Zusammenhang mit ebenen Kurven dritten Grades, Ponceletsche Polygone, geodätische Linien auf dem Umdrehungsellipsoid. Abschnitt II behandelt arithmetische Anwendungen, und zwar zunächst die komplexe Multiplikation und die Klassengleichung. Ein wichtiges Ziel ist hier ein Beweis des Satzes von Abel, demzufolge bei Vorliegen komplexer Multiplikation der „singuläre Modul“ j(ω) durch Wurzelziehen bestimmt werden kann. Ein zweites wesentliches Ziel ist die explizite Berechnung dieser „Klasseninvarianten“ in zahlreichen Beispielen. Weitere zahlentheoretische Anwendungen sind die Berechnung von Darstellungsanzahlen quadratischer Formen und die Bestimmung von Klassenzahlrelationen. – Der (unvollendete) physikalische Teil des Werkes widmet sich ausführlich der wenig bekannten analytischen Theorie des ebenen Gelenkvierecks.

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.

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.

Der mathematische Teil des Werkes beginnt mit numerischen Berechnungen im Gebiet der elliptischen Funktionen. Abschnitt I enthält einen bunten Strauß geometrischer Anwendungen, z.B. Lemniskatenteilung, Zusammenhang mit ebenen Kurven dritten Grades, Ponceletsche Polygone, geodätische Linien auf dem Umdrehungsellipsoid. Abschnitt II behandelt arithmetische Anwendungen, und zwar zunächst die komplexe Multiplikation und die Klassengleichung. Ein wichtiges Ziel ist hier ein Beweis des Satzes von Abel, demzufolge bei Vorliegen komplexer Multiplikation der „singuläre Modul“ j(ω) durch Wurzelziehen bestimmt werden kann. Ein zweites wesentliches Ziel ist die explizite Berechnung dieser „Klasseninvarianten“ in zahlreichen Beispielen. Weitere zahlentheoretische Anwendungen sind die Berechnung von Darstellungsanzahlen quadratischer Formen und die Bestimmung von Klassenzahlrelationen. – Der (unvollendete) physikalische Teil des Werkes widmet sich ausführlich der wenig bekannten analytischen Theorie des ebenen Gelenkvierecks.