Displaying similar documents to “Isomorphisms of algebraic number fields”

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field K with 3 -class group Cl 3 ( K ) of type ( 3 , 3 ) , the structure of the 3 -class groups Cl 3 ( N i ) of the four unramified cyclic cubic extension fields N i , 1 i 4 , of K is calculated with the aid of presentations for the metabelian Galois group G 3 2 ( K ) = Gal ( F 3 2 ( K ) | K ) of the second Hilbert 3 -class field F 3 2 ( K ) of K . In the case of a quadratic base field K = ( D ) it is shown that the structure of the 3 -class groups of the four S 3 -fields N 1 , ... , N 4 frequently determines the type of principalization of the 3 -class group of K in N 1 , ... , N 4 . This...

Unit vector fields on antipodally punctured spheres: big index, big volume

Fabiano G. B. Brito, Pablo M. Chacón, David L. Johnson (2008)

Bulletin de la Société Mathématique de France

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We establish in this paper a lower bound for the volume of a unit vector field v defined on 𝐒 n { ± x } , n = 2 , 3 . This lower bound is related to the sum of the absolute values of the indices of v at x and - x .

A twisted class number formula and Gross's special units over an imaginary quadratic field

Saad El Boukhari (2023)

Czechoslovak Mathematical Journal

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Let F / k be a finite abelian extension of number fields with k imaginary quadratic. Let O F be the ring of integers of F and n 2 a rational integer. We construct a submodule in the higher odd-degree algebraic K -groups of O F using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of F , which is the cardinal of the finite algebraic K -group K 2 n - 2 ( O F ) .

On the 2 -class group of some number fields with large degree

Mohamed Mahmoud Chems-Eddin, Abdelmalek Azizi, Abdelkader Zekhnini (2021)

Archivum Mathematicum

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Let d be an odd square-free integer, m 3 any integer and L m , d : = ( ζ 2 m , d ) . In this paper, we shall determine all the fields L m , d having an odd class number. Furthermore, using the cyclotomic 2 -extensions of some number fields, we compute the rank of the 2 -class group of L m , d whenever the prime divisors of d are congruent to 3 or 5 ( mod 8 ) .

Algebraic independence of the values at algebraic points of a class of functions considered by Mahler

N. Ch. Wass

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This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,..., θ m of complex numbers. Specifically, let K be a number field and let f₁(z),..., f m ( z ) be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) f j ( z b ) = i = 1 m f i ( z ) a i j ( z ) + b j ( z ) (j = i,...,m)for b ≥ 2, a i j ( z ) , b j ( z ) in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the f j ( z ) converge at z = α and the a i j ( z ) , b j ( z ) are analytic at z = α , α b , α b ² , . . . Then the θ i = f i ( α ) are algebraically independent...

The distribution of second p -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

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General concepts and strategies are developed for identifying the isomorphism type of the second p -class group G = Gal ( F p 2 ( K ) | K ) , that is the Galois group of the second Hilbert p -class field F p 2 ( K ) , of a number field K , for a prime p . The isomorphism type determines the position of G on one of the coclass graphs 𝒢 ( p , r ) , r 0 , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field K and of its p -class group Cl p ( K ) , the position of G is restricted to certain admissible branches...

Linear natural operators lifting p -vectors to tensors of type ( q , 0 ) on Weil bundles

Jacek Dębecki (2016)

Czechoslovak Mathematical Journal

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We give a classification of all linear natural operators transforming p -vectors (i.e., skew-symmetric tensor fields of type ( p , 0 ) ) on n -dimensional manifolds M to tensor fields of type ( q , 0 ) on T A M , where T A is a Weil bundle, under the condition that p 1 , n p and n q . The main result of the paper states that, roughly speaking, each linear natural operator lifting p -vectors to tensor fields of type ( q , 0 ) on T A is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

An a b c d theorem over function fields and applications

Pietro Corvaja, Umberto Zannier (2011)

Bulletin de la Société Mathématique de France

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We provide a lower bound for the number of distinct zeros of a sum 1 + u + v for two rational functions u , v , in term of the degree of u , v , which is sharp whenever u , v have few distinct zeros and poles compared to their degree. This sharpens the “ a b c d -theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface x a + y a + z c = 1 contains only finitely many rational or elliptic...

Elements of large order on varieties over prime finite fields

Mei-Chu Chang, Bryce Kerr, Igor E. Shparlinski, Umberto Zannier (2014)

Journal de Théorie des Nombres de Bordeaux

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Let 𝒱 be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C ( 𝒱 ) such that for almost all primes p for all but at most C ( 𝒱 ) points on the reduction of 𝒱 modulo p at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

Bicyclic commutator quotients with one non-elementary component

Daniel Mayer (2023)

Mathematica Bohemica

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For any number field K with non-elementary 3 -class group Cl 3 ( K ) C 3 e × C 3 , e 2 , the punctured capitulation type ϰ ( K ) of K in its unramified cyclic cubic extensions L i , 1 i 4 , is an orbit under the action of S 3 × S 3 . By means of Artin’s reciprocity law, the arithmetical invariant ϰ ( K ) is translated to the punctured transfer kernel type ϰ ( G 2 ) of the automorphism group G 2 = Gal ( F 3 2 ( K ) / K ) of the second Hilbert 3 -class field of K . A classification of finite 3 -groups G with low order and bicyclic commutator quotient G / G ' C 3 e × C 3 , 2 e 6 , according to the algebraic...