LATTICE-ISOMORPHIC GROUPS, AND INFINITE ABELIAN G-COGALOIS FIELD EXTENSIONS

2002 ◽  
Vol 01 (03) ◽  
pp. 243-253 ◽  
Author(s):  
TOMA ALBU ◽  
ŞERBAN BASARAB
Keyword(s):  
Galois Group ◽  
Classical Result ◽  
Infinite Field ◽  
Field Extensions ◽  
Lattices Of Subgroups

The aim of this paper is to provide a proof of the following result claimed by Albu (Infinite field extensions with Galois–Cogalois correspondence (II), Revue Roumaine Math. Pures Appl. 47 (2002), to appear): The Kneser group Kne (E/F) of an Abelian G-Cogalois extension E/F and the group of continuous characters Ch(Gal (E/F)) of its Galois group Gal (E/F) are isomorphic (in a noncanonical way). The proof we give in this paper explains why such an isomorphism is expected, being based on a classical result of Baer (Amer. J. Math.61 (1939), 1–44) devoted to the existence of group isomorphisms arising from lattice isomorphisms of their lattices of subgroups.

Infinite Field Extensions

Algebra ◽  
1991 ◽  
pp. 212-230
Author(s):  
B. L. van der Waerden
Keyword(s):  
Infinite Field ◽  
Field Extensions

scholarly journals Fitting Ideals in Number Theory and Arithmetic

2021 ◽  
Author(s):  
Cornelius Greither
Keyword(s):  
Number Theory ◽  
Galois Group ◽  
Classical Result ◽  
Number Fields ◽  
Class Groups ◽  
Stark Conjecture

AbstractWe describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

scholarly journals Kronecker classes of field extensions of small degree

1991 ◽  
Vol 50 (2) ◽  
pp. 297-315 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Galois Group ◽  
Algebraic Number ◽  
Number Fields ◽  
Small Degree ◽  
Algebraic Number Fields ◽  
Field Extensions

AbstractThe structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.

scholarly journals On a Matrix Representation for Polynomially Recursive Sequences

10.37236/2721 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Christophe Reutenauer
Keyword(s):  
Galois Group ◽  
Matrix Representation ◽  
Field Extensions ◽  
Recursive Sequences ◽  
Canonical Representations

In this article we derive several consequences of a matricial characterization of P-recursive sequences. This characterization leads to canonical representations of these sequences. We show their uniqueness for a given sequence, up to similarity. We study their properties: operations, closed forms, d'Alembertian sequences, field extensions, positivity, extension of the sequence to $\mathbb Z$, difference Galois group.

scholarly journals Construction of semiabelian Galois extensions

1995 ◽  
Vol 37 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Michael Stoll
Keyword(s):  
Galois Group ◽  
Galois Field ◽  
Galois Groups ◽  
Galois Extensions ◽  
Soluble Groups ◽  
Field Extensions ◽  
Galois Field Extensions

This paper shows how to construct Galois field extensions of Hilbertian fields with a given group out of some subclass (called ‘semiabelian groups’ by Matzat [2]) of all soluble groups as Galois group. This is done in a fairly explicit way by constructing polynomials whose Galois groups are universal in the sense that every group in the above subclass is obtained as a quotient of some of them.

scholarly journals LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT -MODULES

2019 ◽  
pp. 1-49 ◽  
Author(s):  
Hui Gao ◽  
Léo Poyeton
Keyword(s):  
Galois Group ◽  
Lie Group ◽  
Galois Representations ◽  
Residue Field ◽  
Field Extensions ◽  
Galois Closure ◽  
Valuation Field

Let $p$ be a prime, let $K$ be a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$ , and let $G_{K}$ be the Galois group. Let $\unicode[STIX]{x1D70B}$ be a fixed uniformizer of $K$ , let $K_{\infty }$ be the extension by adjoining to $K$ a system of compatible $p^{n}$ th roots of $\unicode[STIX]{x1D70B}$ for all $n$ , and let $L$ be the Galois closure of $K_{\infty }$ . Using these field extensions, Caruso constructs the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$ -modules, which classify $p$ -adic Galois representations of $G_{K}$ . In this paper, we study locally analytic vectors in some period rings with respect to the $p$ -adic Lie group $\operatorname{Gal}(L/K)$ , in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$ -modules, we can establish the overconvergence property of the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$ -modules.

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