Kummer Theory over Cyclotomic Zp-extensions

1978 ◽  
pp. 148-165
Author(s):  
Serge Lang
Keyword(s):  
Kummer Theory

Kummer theory for number fields via entanglement groups

2021 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba ◽  
Sebastiano Tronto
Keyword(s):  
Number Fields ◽  
Kummer Theory

scholarly journals Kummer theory for number fields and the reductions of algebraic numbers

2019 ◽  
Vol 15 (08) ◽  
pp. 1617-1633 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Number Field ◽  
Arithmetic Progression ◽  
Direct Proof ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Multiplicative Order ◽  
Higher Rank ◽  
Multiplicative Subgroup ◽  
Torsion Free

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

scholarly journals Explicit Kummer theory for the rational numbers

2020 ◽  
Vol 16 (10) ◽  
pp. 2213-2231
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba ◽  
Sebastiano Tronto
Keyword(s):  
Rational Numbers ◽  
Finitely Generated ◽  
Finite Case ◽  
Kummer Theory ◽  
Multiplicative Subgroup

Let [Formula: see text] be a finitely generated multiplicative subgroup of [Formula: see text] having rank [Formula: see text]. The ratio between [Formula: see text] and the Kummer degree [Formula: see text], where [Formula: see text] divides [Formula: see text], is bounded independently of [Formula: see text] and [Formula: see text]. We prove that there exist integers [Formula: see text] such that the above ratio depends only on [Formula: see text], [Formula: see text], and [Formula: see text]. Our results are very explicit and they yield an algorithm that provides formulas for all the above Kummer degrees (the formulas involve a finite case distinction).

SUR LA CONJECTURE FAIBLE DE GREENBERG DANS LE CAS ABÉLIEN p-DÉCOMPOSÉ

2006 ◽  
Vol 02 (01) ◽  
pp. 49-64 ◽  
Author(s):  
NGUYEN QUANG DO THONG
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Weak Form ◽  
Class Field Theory ◽  
Class Field ◽  
Root Of Unity ◽  
Kummer Theory ◽  
Circular Units

Let p be an odd prime. For any CM number field K containing a primitive pth root of unity, class field theory and Kummer theory put together yield the well known reflection inequality λ+ ≤ λ- between the "plus" and "minus" parts of the λ-invariant of K. Greenberg's classical conjecture predicts the vanishing of λ+. We propose a weak form of this conjecture: λ+ = λ- if and only if λ+ = λ- = 0, and we prove it when K+ is abelian, p is totally split in K+, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case).

scholarly journals Constructing complete tables of quartic fields using Kummer theory

2002 ◽  
Vol 72 (242) ◽  
pp. 941-952 ◽  
Author(s):  
Henri Cohen ◽  
Francisco Diaz y Diaz ◽  
Michel Olivier
Keyword(s):  
Kummer Theory

scholarly journals A note on Kummer theory of division points over singular Drinfeld modules

2001 ◽  
Vol 64 (1) ◽  
pp. 15-20 ◽  
Author(s):  
Anly Li
Keyword(s):  
Number Fields ◽  
Drinfeld Modules ◽  
Kummer Theory ◽  
Complete Analogy

In this paper, we shall establish a Kummer theory of division points over singular Drinfeld modules which is in complete analogy with the classical one in number fields.

scholarly journals A Theorem of Harrison, Kummer Theory, and Galois Algebras

1966 ◽  
Vol 27 (2) ◽  
pp. 663-685 ◽  
Author(s):  
S. U. Chase ◽  
Alex Rosenberg
Keyword(s):  
Galois Group ◽  
Cohomology Group ◽  
Algebraic Closure ◽  
Kummer Theory

Let R be a field and S a separable algebraic closure of R with galois group R. In [8] Harrison succeeded in describing R/′R in terms of R only. More precisely, he constructed a certain complex (R, Q/Z) and proved Homc, where Homc denotes continuous homomorphisms and H2 stands for the second cohomology group of the complex . In this paper, which is mainly expository in nature, we reexamine Harrison’s proof and show how [8] connects with Kummer theory and the theory of galois algebras [16]. We emphasize that most of the ideas on which this paper is based originate in [8].

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