Completeness of the absolute Galois group of the rational number field

1977 ◽  
Vol 1977 (291) ◽  
pp. 1-22
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Number ◽  
Absolute Galois Group ◽  
The Absolute

scholarly journals On nilpotent extensions of algebraic number fields I

1992 ◽  
Vol 125 ◽  
pp. 1-14
Author(s):  
Katsuya Miyake ◽  
Hans Opolka
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Number Fields ◽  
Abelian Extension ◽  
Central Series ◽  
Schur Multiplier ◽  
Absolute Galois Group ◽  
Algebraic Number Fields ◽  
The Absolute

The lower central series of the absolute Galois group of a field is obtained by iterating the process of forming the maximal central extension of the maximal nilpotent extension of a given class, starting with the maximal abelian extension. The purpose of this paper is to give a cohomological description of this central series in case of an algebraic number field. This description is based on a result of Tate which states that the Schur multiplier of the absolute Galois group of a number field is trivial. We are in a profinite situation throughout which requires some functorial background especially for treating the dual of the Schur multiplier of a profinite group. In a future paper we plan to apply our results to construct a nilpotent reciprocity map.

scholarly journals The Absolute Galois Group of the Field of Totally S-Adic Numbers

2009 ◽  
Vol 194 ◽  
pp. 91-147 ◽  
Author(s):  
Dan Haran ◽  
Moshe Jarden ◽  
Florian Pop
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Free Product ◽  
Absolute Galois Group ◽  
Fixed Field ◽  
Local Factors ◽  
Finite Set ◽  
The Absolute

AbstractFor a finite set S of primes of a number field K and for σ1,…, σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1,…,σe in Ktot,S by Ktot,S(σ). We prove that foralmost all σ ∈ Gal(K)e the absolute Galois group of Ktot,S(σ) is the free product of and a free product of local factors over S.

scholarly journals p-adic Uniformization and the Action of Galois on Certain Affine Correspondences

2018 ◽  
Vol 61 (3) ◽  
pp. 531-542
Author(s):  
Patrick Ingram
Keyword(s):  
Dynamical Systems ◽  
Galois Group ◽  
Number Field ◽  
Directed Graph ◽  
Absolute Galois Group ◽  
Polynomial Dynamical Systems ◽  
The Absolute ◽  
Holomorphic Correspondences

AbstractGiven two monic polynomials ƒ and g with coefficients in a number field K, and some ∞ ∈ K, we examine the action of the absolute Galois group Gal(/K) on the directed graph of iterated preimages of ∞ under the correspondence g(y) = ƒ(x), assuming that deg(ƒ) > deg(g) and that gcd (deg(ƒ), deg(g)) = 1. If a prime of K exists at which ƒ and g have integral coefficients and at which ∞ is not integral, we show that this directed graph of preimages consists of finitely many Gal(/K)-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Böttcher’s uniformization of polynomial dynamical systems over , although the construction of a Böttcher coordinate for complex holomorphic correspondences remains unresolved.

scholarly journals The Sylow subgroups of the absolute Galois group Gal(Q)

2015 ◽  
Vol 284 ◽  
pp. 186-212 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Moshe Jarden ◽  
Danny Neftin
Keyword(s):  
Galois Group ◽  
Absolute Galois Group ◽  
Sylow Subgroups ◽  
The Absolute

Zeta functions of twisted modular curves

2006 ◽  
Vol 80 (1) ◽  
pp. 89-103 ◽  
Author(s):  
Cristian Virdol
Keyword(s):  
Zeta Function ◽  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Modular Curves ◽  
Absolute Galois Group ◽  
Modular Curve ◽  
The Absolute

AbstractIn this paper we compute and continue meromorphically to the whole complex plane the zeta function for twisted modular curves. The twist of the modular curve is done by a modprepresentation of the absolute Galois group.

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

1979 ◽  
Vol 75 ◽  
pp. 121-131 ◽  
Author(s):  
Susumu Shirai
Keyword(s):  
Nilpotent Group ◽  
Galois Group ◽  
Number Field ◽  
Direct Product ◽  
Rational Number ◽  
Nilpotency Class ◽  
Abelian Extension ◽  
Galois Groups ◽  
Classification Theory ◽  
Ray Class Field

Let Q be the rational number field, K/Q be a maximal Abelian extension whose degree is some power of a prime l, and let f(K) be the conductor of K/Q; if l = 2, let K be complex, and if in addition f(K) ≡ 0 (mod 2), let f(K) ≡ 0 (mod 16). Denote by (K) the Geschlechtermodul of K over Q and by K̂ the maximal central l-extension of K/Q contained in the ray class field mod (K) of K. A. Fröhlich [1, Theorem 4] completely determined the Galois group of K̂ over Q in purely rational terms. The proof is based on [1, Theorem 3], though he did not write the proof in the case f(K) ≡ 0 (mod 16). Moreover he gave a classification theory of all class two extensions over Q whose degree is a power of l. Hence we know the set of fields of nilpotency class two over Q, because a finite nilpotent group is a direct product of all its Sylow subgroups. But the theory becomes cumbersome, and it is desirable to reconstruct a more elementary one.

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