Algebraic number fields (Q1090365)

scientific article; zbMATH DE number 4006368

Language	Label	Description	Also known as
English	Algebraic number fields	scientific article; zbMATH DE number 4006368

Statements

instance of

scholarly article

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title

Algebraic number fields (English)

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author

L. V. Kuz'min

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published in

Journal of Soviet Mathematics

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publication date

1987

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review text

Translation from Itogi Nauki Tekh., Ser. Algebra Topol. Geom. 22, 117--204 (Russian) (1984; Zbl 0563.12002).

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zbMATH Keywords

algebraic numbers

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arithmetic of abelian fields

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bibliography

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survey

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local fields

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cyclotomic fields

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Gamma-extensions

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Galois theory

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reference list

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p-adic L-functions

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MaRDI profile type

Publication

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cites work

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Class numbers in $\mathbb{Z}_p^d$-extensions

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Computing the number of totally positive circular units which are squares

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Periodic expansions and units in quadratic and cubic number fields.

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The p-Adic Gamma Measures

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On Some Families of Imaginary Quadratic Fields

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Quadratic Fields With 3-Rank Equal to 4

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Some contributions to the theory of cyclic quartic extensions of the rationals

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Euclid's algorithm in pure quartic fields

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Computation of the 2-Rank of Pure Cubic Fields

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Polynomials with PSL(2,7) as Galois group

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Fields generated by linear combinations of roots of unity, II.

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Q5680219

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Regulators and Iwasawa modules

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Some lattices over $Q(\sqrt{-3})^*$

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Iwasawa invariants of abelian number fields

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The Iwasawa invariant $\mu_p$ vanishes for abelian number fields

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Independent systems of units in certain algebraic number fields.

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Principal ideal theorems in the genus field for absolutely abelian extensions

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G-closed fields and imbeddings of quadratic number fields

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Local trace forms

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Extensions of the Hasse norm theorem

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Extensions of the hasse norm theorem, II

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Unit signatures, and even class numbers, and relative class numbers.

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Units with norm - 1 and signatures of units.

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Partially normal radical extensions of the rationals

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Artin Root Numbers for Real Characters

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Ranks of Sylow 3-subgroups of ideal class groups of certain cubic fields

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Number Fields with Prescribed l-Class Groups

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Cubic fields whose class numbers are not divisible by 3

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Structure of l ‐class groups of certain number fields and ℤ l ‐extensions

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The Hasse norm principle for Abelian extensions of number fields

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The Hasse norm principle in cyclotomic number fields.

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Q4192153

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The Ideal Class Groups of Two Cyclotomic Fields

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Galois groups of intersections of local fields

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Plongement d'une extension d'ordre p ou p2 dans une surextension non abelienne d'ordre p3.

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Q4101935

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Formulations de la conjecture de Leopoldt et étude d'une condition suffisante

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Unités cyclotomiques, unités semi-locales et $\mathbb{Z}_\ell$-extensions

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Remarques sur les unites cyclotomiques et les unites elliptiques

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Unites elliptiques et unites cyclotomiques

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Q3890795

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Groupes d'unités elliptiques

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Elementare Berechnung von kubischen Diskriminanten

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Q4175654

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Real characters of the ideal class-group and the narrow ideal class-group of Q(√d)

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$\Gamma$-extensions of imaginary quadratic fields

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$\Gamma$-extensions of imaginary quadratic fields. II

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The nontriviality of certain Z$_l$-extensions

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Examples of Iwasawa invariants

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Examples of Iwasawa invariants, II

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Genera in Abelian Extensions

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$\mathbb{Z}_3$-invariants of real and imaginary quadratic fields

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Q4159455

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Hilbert class fields and split extensions

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Genera in normal extensions

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Some Applications of ABHYANKAR's Lemma

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The admissibility of $A_5$

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Sur les $l$-classes d'ideaux des extensions non galoisiennes de Q de degré premier impair $l$ à cloture galoisienne diedrale de degré 2$l$

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Critère de parité du nombre de classes des extensions abéliennes réelles de $Q$ de degré impair

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Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés

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Annulation du groupe des $\ell$-classes généralisées d'une extension abélienne réelle de degré premier à $\ell$

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Signature des unités cyclotomiques et parité du nombre de classes des extensions cycliques de $Q$ de degré premier impair

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Classes and units of real cyclic extensions of degree 4 of $\mathbb{Q}$

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On a certain $l$-adic representation

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On p-adic L-functions and cyclotomic fields

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On the Iwasawa Invariants of Totally Real Number Fields

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Onp-adicL-functions and cyclotomic fields. II

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On the structure of certain Galois groups

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A Note on K 2 and the Theory of ℤ p -Extensions

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On 2-adic L-functions and cyclotomic invariants

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Explizite Formeln für Einheiten algebraischer Zahlkörper und eine Familie irreduzibler Polynome über Funktionenkörpern

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Gauss sums and the p-adic $\Gamma$-function

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Units and discriminants of algebraic number fields

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Consecutive integers in algebraic number fields.

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Ideal-theoretic characterization of the relative genus field.

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The Hasse norm principle in non-abelian extensions.

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Unabhängige Einheitensysteme für eine allgemeine Klasse algebraischer Zahlkörper

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On 3-class groups of non-Galois cubic fields

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On the group automorphisms of the absolute Galois group of the rational number field

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Completeness of the absolute Galois group of the rational number field

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Completeness of the absolute Galois group of the rational number field

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Some unramified abelian extensions of algebraic number fields.

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The genus fields of algebraic number fields

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A note on the law of decomposition of primes in certain Galois extension

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Algebraic extensions of finite corank of Hilbertian fields

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On Cebotarev sets

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An analogue of Artin-Schreier theorem

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On the characterization of local fields by their absolute Galois groups

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Determining the Fundamental Unit of a Pure Cubic Field Given Any Unit

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Complete determinations of the 3-class rank in pure cubic fields

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A class of fundamental units and some classes of Jacobi-Perron algorithms in pure cubic fields

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On the integral basis of the maximal real subfield of a cyclotomic field.

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The minimum discriminant of sixth degree totally complex algebraic number fields

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On p-adic L-functions associated to elliptic curves

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Steinitz Classes of Cyclic Extensions of Degree l r

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Units of modulus 1

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A note on the normality of unramified, Abelian extensions of quadratic extensions

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A note on Friedlander's paper "On the class numbers of certain quadratic extensions"

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On normal radical extensions of the rationals

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Prime ideal decomposition in $F({\root m\of\mu})$

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Euclidean rings of algebraic numbers and functions

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Q4132638

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Tours de corps de classes et estimations de discriminants

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Small discriminants

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On Euclidean rings of integers in cyclotomic fields.

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Solution of the class number two problem for cyclotomic fields

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Q3864603

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On the first factor of the class number of prime cyclotomic fields

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Q4101931

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Q3861511

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ON THE CLASS FIELD TOWER OF AN IMAGINARY QUADRATIC NUMBER FIELD

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Konstruktion von Zahlkörpern mit der Galoisgruppe $M_{11}$ über $\mathbb{Q}(\sqrt{-11})$

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A matrix criterion for normal integral bases

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Invariants of the trace-form of a number field

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Arithmetic properties of the idéle discriminant

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Notes on étale cohomology of number fields

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Some Remarks on Computing Galois Groups

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On the Galois cohomology of Dedekind rings

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Class Numbers and μ-Invariants of Cyclotomic Fields

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On the Iwasawa invariants of imaginary abelian fields

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On the cyclotomic invariants of Iwasawa.

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Iwasawa invariants and Kummer congruences

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Q4094956

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Konvexe Untergruppen in algebraischen Zahlkörpern

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On the genus of norm forms

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On some Galois cohomology groups of a local field and its application to the maximal p-extension

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A note on Maus' theorem on ramification groups

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On the maximal abelianℓ-extension of a finite algebraic number field with given ramification

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On Grunwald-Hasse-Wang's theorem

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Q4141307

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A normal integral basis theorem for dihedral groups

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On the module structure of the ring of all integers of a p-adic number field

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On the module structure in a cyclic extension over a -adic number field

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On the module structure of a p-extension over a -adic number field

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Q5669400

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On p-adic power series

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Some invariants of $\mathbb{Z}_p^d$

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Real quadratic fields with large class number

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On units of certain cubic number fields

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On the radius of convergence of the p-adic L-function

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Sommes de carrés d'entiers d'un corps $p$-adique

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Units and class numbers of a dihedral Galois extension of $\mathbb{Q}$

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ON BIQUADRATIC FIELDS

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Verallgemeinerung eines Satzes von Rabinowitsch über imaginär-quadratische Zahlkörper.

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Q4186440

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An explicit formula for the fundamental units of a real pure sextic number field and its Galois closure

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On the 3-Rank of Quadratic Fields and the Euler Product

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Einheiten in algebraischen Funktionen- und Zahlkörpern.

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Über das Einbettungsproblem der algebraischen Zahlentheorie

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Einbettungsprobleme mit lokaler Vorgabe und freie Produkte lokaler Galoisgruppen.

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Eine Bemerkung zum Existenzsatz von Grunwald-Hasse-Wang.

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Q4081367

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Zur Theorie der nilpotenten algebraischen Zahlkörper

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On solvable number fields

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Über das Einbettungsproblem für globale Körper bei beschränkter Verzweigung

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Einige Klassen von endlichen GALOIS-Moduln mit HASSE-Prinzip

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Ein Endlichkeitssatz der Galois-Kohomologie

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Proper solutions of the imbedding problem with restricted ramification

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Q4047637

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Explizite Darstellungen von Einheiten und ihre Anwendung auf Mehrklassigkeitsfragen bei reell-quadratischen Zahlkörpern. II.

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Some analytic estimates of class numbers and discriminants

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Lower bounds for discriminants of number fields

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Lower bounds for discriminants of number fields. II

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On the norms of algebraic integers

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Euclid's Algorithm in the Cyclotomic Field Q(ζ 16 )

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Q4124916

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Q4190717

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Quadratic fields and Hopf fibrations

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Q4187899

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On some truncated units in algebraic number fields of degree $n> 3$

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On Dirichlet's Class Number Formula

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Relation entre les 2-groupes des classes d'idéaux au sens ordinaire et restreint de certains corps de nombres

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Rélations entre les 2-groupes des classes d'idéaux des extensions quadratiques $k(\sqrt d)$ et $k(\sqrt{-d})$

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Sur la divisibilite par 8 et 16 des nombres de classes d'ideaux des corps quadratiques $Q(\sqrt{2p})$ et $Q(\sqrt{-2p})$

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Computations on the growth of the first factor for prime cyclotomic fields

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Computations on the growth of the first factor for prime cyclotomic fields II

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Pure quartic number fields whose class numbers are even.

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Class number relations in pure sextic fields.

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Units of algebraic numberfields

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On a conjecture of Brandler

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Class number formulae for bicubic fields

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A genus theory for quartic fields.

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The class number of pure fields of prime degree

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The class number of pure fields of prime degree

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On the equation $\zeta_K(s)=\zeta_{K'}(s)$

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Die Einseinheitengruppen höherer Stufen einer zerfallenden zahm‒verzweigten Erweiterung als G<scp>ALOIS</scp>moduln

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Steinitz Classes in Quartic Fields

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On the 2-part of the class number of imaginary quadratic number fields

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Berechnung kleiner Diskriminanten total reeller algebraischer Zahlkörper.

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Über biquadratische Zahlkörper gleicher Diskriminante

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Invarianten des total reellen Körpers siebten Grades mit Minimaldiskriminante

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Regulatorabschätzungen für total reelle algebraische Zahlkörper

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An Effective Number Geometric Method of Computing the Fundamental Units of an Algebraic Number Field

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Q3859890

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Minorations de discriminants [d’après A. M. Odlyzko]

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Arithmetic euclidean rings

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Q4154657

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Über die gruppentheoretische Struktur der Relationen zwischen Relativnormabbildungen in endlichen Galoisschen Körpererweiterungen

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Q4777400

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Q3904661

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A modular construction of unramified p-extensions of $\mathbb{Q}(\mu_p)$

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Q4137253

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Q4195048

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On the Order and Degree of Solutions to Pure Equations

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Über eine Erweiterung der Wittschen Reduktion bei der Bestimmung der Schurindizes von endlichen Gruppen über lokalen Körpern

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$\mathfrak p$-adic fields having the same type of algebraic extensions

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Q3892333

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Q4073459

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Nombres de Hurwitz et unités elliptiques. Un critère de régularité pour les extensions abéliennes d'un corps quadratique imaginaire

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The non-vanishing of certain Hecke $L$-functions at the center of the critical strip

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A generalization of Berwick's unit algorithm

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Q4094958

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Zur Existenz einfacher abelscher Varietäten mit komplexer Multiplikation.

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On the Pythagoras number of orders in totally real number fields.

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Arithmetische Ausdeutung der Klassenzahlformel für einfach reelle kubische Zahlkörper

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Über die Klassenzahl gewisser nicht galoisscher Körper 6-ten Grades

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Die Klassenzahl der Teilkörper abelscher Erweiterungen imaginär-quadratischer Zahlkörper. I.

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Die Klassenzahl der Teilkörper abelscher Erweiterungen imaginär-quadratischer Zahlkörper. II.

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Zur Theorie der Ringklassenkörper über imaginär-quadratischen Zahlkörpern

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Q4156853

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Über die analytische Klassenzahlformel für reelle abelsche Zahlkörper.

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Eine Abschätzung der Klassenzahl gewisser reiner Zahlkörper sechsten Grades.

0 references

Die Relationen von Gaußschen Summen und Kreiseinheiten

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Größencharaktere und Relativklassenzahl abelscher Zahlkörper

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Über gewisse Galoiscohomologiegruppen

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Über die Werte der Riemannschen Zetafunktion an den ganzzahligen Stellen.

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Ramification in p-adic Lie extensions

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Lie algebras of Galois groups arising from Hodge-Tate modules

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On explicit reciprocity laws.

0 references

On explicit reciprocity laws. II.

0 references

Q4168691

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Q4175668

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Units in totally complex $S_3$ fields

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New types of quadratic fields having three invariants divisible by 3

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The Simplest Cubic Fields

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Profinite Groups, Arithmetic, and Geometry. (AM-67)

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Q4119293

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On certain ray class invariants of real quadratic fields

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On certain ray class invariants of real quadratic fields

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Central class numbers in central class field towers

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On the central class field mod of Galois extensions of an algebraic number field

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A remark concerning the 2-adic number field

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On Galois groups of class two extensions over the rational number field

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On the central ideal class group of cyclotomic fields

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ON CERTAIN VALUES OF p-ADIC L-FUNCTIONS

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Q4135674

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On relative class numbers of certain quadratic extensions

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On the norm form of a finite Galois extension over 𝑄

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Class numbers of binary quadratic forms over algebraic number fields.

0 references

Q4123411

0 references

On the Stickelberger ideal and the circular units of a cyclotomic field

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On the Stickelberger ideal and the circular units of an abelian field

0 references

Q3874311

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Primes which remain irreducible in a normal field

0 references

Q4117911

0 references

Normal bases for quadratic extensions

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Zeta functions and integral representation theory

0 references

On the embedding problem for nonsolvable Galois groups of algebraic number fields: Reduction theorems

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Localizability of the embedding problem with symplectic kernel

0 references

Frobenius Galois groups over quadratic fields

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Class groups and Brauer groups

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SL(2,5) and Frobenius Galois Groups Over Q

0 references

K-théorie des anneaux d'entiers de corps de nombres et cohomologie etale

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On Complex Quadratic Fields with Class-Number Two

0 references

The analytic theory of algebraic numbers

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$L$-functions at $s=1$. II: Artin $L$-functions with rational characters

0 references

$L$-functions at $s=1$. III: Totally real fields and Hilbert's twelfth problem

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$L$-functions at $s=1$. IV: First derivatives at $s=0$

0 references

The genus theory of number fields

0 references

Hilbert’s twelfth problem and 𝐿-series

0 references

Q4137272

0 references

On an Algorithm of Billevich for Finding Units in Algebraic Number Fields

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Über die Einheitengruppe der reinen algebraischen Zahlkörper sechsten Grades.

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Eine Formel für Grundeinheiten in reinen algebraischen Zahlkörpern dritten, vierten und sechsten Grades

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``Verstümmelte'' Grundeinheiten für biquadratische und bikubische Zahlkörper

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Über die Grundeinheit der reell-quadratischen Zahlkörper Q( A2N2 + BN + C).

0 references

Zur Parametisierung reell-quadratischer Zahlkörper.

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ON RAMIFICATION OF p-EXTENSIONS OF $p$-ADIC NUMBER FIELDS

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Q3853734

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The existence of natural field structures for finite dimensional vector spaces over local fields

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Relations between $K_2$ and Galois cohomology

0 references

Q4196515

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Q4163310

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Q4161408

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Norms from quadratic fields and their relation to noncommuting $2\times 2$ matrices. III: A link between the 4-rank of ideal class groups in $Q(\sqrt m)$ and in $Q(\sqrt{-m})$.

0 references

Galois module structure of classgroups and units

0 references

On the self-duality of a ring of integers as a Galois module

0 references

Galois module structure of integers of relative abelian extensions.

0 references

Adams Operations, Local Root Numbers, and the Galois Module Structure of Rings of Integers

0 references

Galois module structure of rings of integers

0 references

A logarithmic approach to classgroups of integral group rings

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Galois Module Structure of Rings of Integers in Kummer Extensions

0 references

On Fröhlich's conjecture for rings of integers of tame extensions

0 references

Q3863959

0 references

Q4190720

0 references

Berechnung von Einheiten nach Lagrange.

0 references

On the local Langlands conjecture for GL(2)

0 references

Class numbers of cubic cyclic fields

0 references

On Artin L-functions

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On a cubic field with discriminant $163^2$

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Isomorphisms of Galois groups

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Isomorphisms of Galois groups of solvably closed Galois extensions

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Separably Hilbertian fields

0 references

VANDIVER'S CONGRUENCE FOR THE RELATIVE CLASS NUMBER OF AN IMAGINARY ABELIAN FIELD

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A characterization of completely regular fields

0 references

Prime ideal decomposition in $F(\mu^{1/p})$

0 references

Transcendance et exponentielles en plusieurs variables

0 references

The ambiguous class group and the genius group of certain non‐normal extensions

0 references

Kuroda's class number relation

0 references

Class numbers and $\mathbb Z_p$-extensions

0 references

A note on p-adic L-functions

0 references

Relative Integral Bases

0 references

On Fermat's last theorem.

0 references

The calculation of $L_p(1,\chi)$

0 references

The non-p-part of the class number in a cyclotomic $\mathbb{Z}_p$-extension

0 references

Units of irregular cyclotomic fields

0 references

An arithmetical application of elliptic functions to the theory of cubic residues

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On class numbers of some cyclotomic fields.

0 references

Pieces of eight in class groups of quadratic fields

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A Nonsymmetric Hasse-Minkowski Theorem

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A Probable Hasse Principle for Pencils of Quadrics

0 references

Q4143407

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Exercices dyadiques

0 references

Higher explicit reciprocity laws

0 references

Modular curves and the class group of $\mathbb{Q} (\zeta_p)$

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Certain Pure Cubic Fields With Class-Number One

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Calculation of the Regulator of Q(√D) by use of the Nearest Integer Continued Fraction Algorithm

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A Note on Class-Number One in Pure Cubic Field

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The class number of $\mathbb Q(\sqrt{-p})$ modulo 4, for $p\equiv 3$ (mod 4) a prime

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Some counter-examples in the theory of the Galois module structure of wild extensions

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Die Einseinheitengruppe von p-Erweiterungen regulärer -adischer Zahlkörper als Galoismodul.

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Existenz unendlicher algebraischer Zahlkörper, über denen jedes Einbettungsproblem lösbar ist

0 references

Q3864611

0 references

Q3048922

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A purely algebraic proof of special cases of Tchebotarev's theorem

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What is a Reciprocity Law?

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Construction of number fields with prescribed $\ell$-class groups

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Schur indices over the 2-adic field

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A counterexample for the local analogy of a theorem by lwasawa and Uchida

0 references

On the class-number of the maximal real subfield of a cyclotomic field.

0 references

A note on formal groups and zeta functions.

0 references

A Kronecker limit formula for real quadratic fields

0 references

On the values at negative integers of the zeta-function of a real quadratic field

0 references

Reduction theory in algebraic number fields

0 references

Zum Hauptidealsatz von Tannaka‐Terada

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Ergänzungen zu Weils Exercises dyadiques

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Weil-Darstellungen und lokale Galoistheorie

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Identifiers

zbMATH Open document ID

0621.12002

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DOI

10.1007/BF01093434

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Mathematics Subject Classification ID

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Sitelinks

Mathematics(1 entry)

mardi Publication:1090365