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Title Details:
Basic Galois Theory
Authors: Marmaridis, Nikolaos-Theodosios
Subject: MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > GENERAL ALGEBRAIC SYSTEMS > ALGEBRAIC STRUCTURES
MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > NUMBER THEORY > FINITE FIELDS AND COMMUTATIVE RINGS (NUMBER-THEORETIC ASPECTS)
MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > FIELD THEORY AND POLYNOMIALS
MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > FIELD THEORY AND POLYNOMIALS > GENERAL FIELD THEORY
MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > FIELD THEORY AND POLYNOMIALS > FIELD EXTENSIONS
MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > REAL FUNCTIONS > POLYNOMIALS, RATIONAL FUNCTIONS
Keywords:
Divisibility
Unique Factorization Domains
Gauss Lemma
Field Extensions
Algebraic Extensions
Separable and Normal Extensions
Galois Extensions
Symmetric Polynomials
Fundamental Theorem of Galois Theory
Finite Fields
Geometric Constructions
Galois Groups of Polynomials
Solvability by Radicals
Relative Resolvents
Resolvent Weber
Description:
Abstract:
This book is a thorough presentation of Galois Theory. After recalling basic notions of ring theory, we prove Gauss Lemma for unique factorization domais. We also present and prove a generalization of Eisenstein’s Criterion for integral domains. We study algebraic, separable and normal extensions of fields. We prove the existence of algebraicaly closed fields and of the algebraic hull of a field. We characterize Galois extensions via the theory of group actions on sets and we also study finite Galois extensions. We prove Artin’s Lemma and present a procedure for determining fixed fields (relative to a subgroup of the Galois group) of simple finite field extensions. We study the field of rational functions as an extension of the field of symmetric rational functions. Using the theory of symmetric polynomials, we obtain the formulas describing the roots of the cubic and biquadratic polynomials. We prove the Fundamental Theorem of Galois Theory and the Fundamental Theorem of Algebra. We study extensively the finite fields. We give and prove necessary and sufficient conditions in order to be a complex number constructible by compass and lineal. We also give and prove necessary and sufficient conditions in order to be a regular n-gon constructible by compass and lineal. We study the splitting fields of cyclotomic polynomials and of polynomials of the form x^n-a. We prove Galois Theorem of the solvability of a polynomial by radicals. We determine the Galois groups of cubic and biquadratic polynomials. Using relative resolvents, we study the the Galois group of a polynomial. Finally we present and study the Weber resolvent which gives a sufficient and necessary condition in order to be solvable by radicals an irreducible polynomial of fifth degree with rational coefficients. The book contais many non trivial examples and over 365 exercises.
Technical Editors: Marmaridis, Nikolaos-Theodosios
Type: Undergraduate textbook
Creation Date: 2021
Item Details:
ISBN 978-618-85370-2-6
License: Attribution - NonCommercial - ShareAlike 4.0 International (CC BY-NC-SA 4.0)
DOI http://dx.doi.org/10.57713/kallipos-7
Handle http://hdl.handle.net/11419/8006
Bibliographic Citation: Marmaridis, N. (2021). Basic Galois Theory [Undergraduate textbook]. Kallipos, Open Academic Editions. http://dx.doi.org/10.57713/kallipos-7
Language: Greek
Consists of:
1. Rings and Polynomials
2. Divisibility
3. Unique Factorization Domains and Polynomial Rings
4. Field Extensions
5. Algebraic Field Extensions
6. Separable Extensions, Normal Extensions
7. Galois Extensions
8. Finite Galois Extensions
9. Symmetric Polynomials and Applications
10. Fundamental Galois Theorem
11. Galois Groups and Polynomials
12. Geometric Constructions and Galois Theory
13. Galois Groups of Polynomials as Subgroups of Symmetric Groups
Publication Origin: Kallipos, Open Academic Editions