國立虎尾科技大學 |

Galois theories of fields and rings

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Galois theories of fields and rings/ by Francis Borceux.
Author:	Borceux, Francis.
Published:	Cham :Springer Nature Switzerland : : 2024.,
Description:	xii, 181 p. :ill., digital ; : 24 cm.;
Contained By:	Springer Nature eBook
Subject:	Galois theory. -
Online resource:	https://doi.org/10.1007/978-3-031-58460-2
ISBN:	9783031584602

Galois theories of fields and rings
Borceux, Francis.

Galois theories of fields and rings[electronic resource] /by Francis Borceux. - Cham :Springer Nature Switzerland :2024. - xii, 181 p. :ill., digital ;24 cm. - Coimbra mathematical texts,v. 22813-0065 ;. - Coimbra mathematical texts ;v. 2..

Historical introduction -- Part I Some Galois theorems for fields -- 1 The classical Galois theorem -- 2 The Galois theorem of Grothendieck -- 3 Profinite topological spaces -- 4 The Galois theorems in arbitrary dimension -- Part II The Galois theory of rings -- 5 Adjunctions and monads -- 6 Profinite groupoids and presheaves -- 7 The descent theory of rings -- 8 The Pierce spectrum of a ring -- 9 The Galois theorem for rings -- Further Reading -- Index.

This textbook arises from a master's course taught by the author at the University of Coimbra. It takes the reader from the very classical Galois theorem for fields to its generalization to the case of rings. Given a finite-dimensional Galois extension of fields, the classical bijection between the intermediate field extensions and the subgroups of the corresponding Galois group was extended by Grothendieck as an equivalence between finite-dimensional split algebras and finite sets on which the Galois group acts. Adding further profinite topologies on the Galois group and the sets on which it acts, these two theorems become valid in arbitrary dimension. Taking advantage of the power of category theory, the second part of the book generalizes this most general Galois theorem for fields to the case of commutative rings. This book should be of interest to field theorists and ring theorists wanting to discover new techniques which make it possible to liberate Galois theory from its traditional restricted context of field theory. It should also be of great interest to category theorists who want to apply their everyday techniques to produce deep results in other domains of mathematics.

ISBN: 9783031584602

Standard No.: 10.1007/978-3-031-58460-2doiSubjects--Topical Terms:

677326
Galois theory.

LC Class. No.: QA214

Dewey Class. No.: 512.32

Galois theories of fields and rings
LDR:02706nam a22003495a 4500 001 1135204
003 DE-He213
005 20240820130242.0
006 m d
007 cr nn 008maaau
008 241213s2024 sz s 0 eng d
020 $a 9783031584602 $q (electronic bk.)
020 $a 9783031584596 $q (paper)
024 7 $a 10.1007/978-3-031-58460-2 $2 doi
035 $a 978-3-031-58460-2
040 $a GP $c GP
041 0 $a eng
050 4 $a QA214
072 7 $a PBF $2 bicssc
072 7 $a MAT002000 $2 bisacsh
072 7 $a PBF $2 thema
082 0 4 $a 512.32 $2 23
090 $a QA214 $b .B726 2024
100 1 $a Borceux, Francis. $3 1021389
245 1 0 $a Galois theories of fields and rings $h [electronic resource] / $c by Francis Borceux.
260 $a Cham : $c 2024. $b Springer Nature Switzerland : $b Imprint: Springer,
300 $a xii, 181 p. : $b ill., digital ; $c 24 cm.
347 $a text file $b PDF $2 rda
490 1 $a Coimbra mathematical texts, $x 2813-0065 ; $v v. 2
505 0 $a Historical introduction -- Part I Some Galois theorems for fields -- 1 The classical Galois theorem -- 2 The Galois theorem of Grothendieck -- 3 Profinite topological spaces -- 4 The Galois theorems in arbitrary dimension -- Part II The Galois theory of rings -- 5 Adjunctions and monads -- 6 Profinite groupoids and presheaves -- 7 The descent theory of rings -- 8 The Pierce spectrum of a ring -- 9 The Galois theorem for rings -- Further Reading -- Index.
520 $a This textbook arises from a master's course taught by the author at the University of Coimbra. It takes the reader from the very classical Galois theorem for fields to its generalization to the case of rings. Given a finite-dimensional Galois extension of fields, the classical bijection between the intermediate field extensions and the subgroups of the corresponding Galois group was extended by Grothendieck as an equivalence between finite-dimensional split algebras and finite sets on which the Galois group acts. Adding further profinite topologies on the Galois group and the sets on which it acts, these two theorems become valid in arbitrary dimension. Taking advantage of the power of category theory, the second part of the book generalizes this most general Galois theorem for fields to the case of commutative rings. This book should be of interest to field theorists and ring theorists wanting to discover new techniques which make it possible to liberate Galois theory from its traditional restricted context of field theory. It should also be of great interest to category theorists who want to apply their everyday techniques to produce deep results in other domains of mathematics.
650 0 $a Galois theory. $3 677326
650 0 $a Algebraic fields. $3 672441
650 0 $a Rings (Algebra) $3 786669
650 1 4 $a Algebra. $2 gtt $3 579870
650 2 4 $a Mathematics. $3 527692
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
830 0 $a Coimbra mathematical texts ; $v v. 2. $3 1456853
856 4 0 $u https://doi.org/10.1007/978-3-031-58460-2
950 $a Mathematics and Statistics (SpringerNature-11649)