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This textbook provides an undergraduate introduction to Galois theory and its most notable applications.
Galois theory was born in the 19th century to study polynomial equations. Both powerful and elegant, this theory was at the origin of a substantial part of modern algebra and has since undergone considerable development. It remains an extremely active research subject and has found numerous applications beyond pure mathematics. In this book, the authors introduce Galois theory from a contemporary point of view. In particular, modern methods such as reduction modulo prime numbers and finite fields are introduced and put to use. Beyond the usual applications of ruler and compass constructions and solvability by radicals, the book also includes topics such as the transcendence of e and π, the inverse Galois problem, and infinite Galois theory.
Based on courses of the authors at the École Polytechnique, the book is aimed at students with a standard undergraduate background in (mostly linear) algebra. It includes a collection of exam questions in the form of review exercises, with detailed solutions.
Preface
Prologue
Invitation to Galois Theory
Construction With a Straightedge and Compass
Solving Equations
Basic Concepts of Group Theory
Groups
Quotient Groups
Supplement on Commutative Groups
Exact Sequences
Group Actions
Symmetric Groups
Solvable Groups
Basic Concepts of Ring Theory
Rings
Rings of Polynomials
Field Morphisms
Quotient Rings
The Characteristic
Domains and Properties of Ideals
The Rank of a Finite Type Free Module
The Chinese Lemma
The Frobenius Morphism
Basic Concepts of Algebras Over a Field
Algebras and Algebra Morphisms
The Degree of an Algebra
Rupture Fields
Algebraic and Transcendental Elements
The Degree of Transcendence
Algebraicity Criteria
The Concept of Algebraic Closure
Proof of the Existence of the Algebraic Closure
Proof of the Uniqueness of the Algebraic Closure
The Splitting Field of a Polynomial
Finite Fields and Perfect Fields
Existence and Uniqueness of Finite Fields
Automorphisms of Finite Fields
An Application of the Chinese Lemma: The Berlekamp Algorithm
Where We Reduce to P Without Square Factor
Fixed Points of the Frobenius Morphism
Factorization of P
Extensions of Perfect Fields
Separable Polynomials and Perfect Fields
The Primitive Element Theorem
The Galois Correspondence
Galois Extensions
Characterizations of Galois Extensions
The Galois Group of Finite Fields
Fixed Points
Statement and Proof of the Galois Correspondence
Addendum: Infinite Galois Correspondence
Topology of the Galois Group
Infinite Galois Correspondence
Cyclotomy and Constructibility
Cyclotomic Extensions
On the Galois Group of the General Cyclotomic Extension
Irreducibility of the Cyclotomic Polynomial Over Q
Intersections of Cyclotomic Fields
Constructibility With a Straightedge and Compass
Solvability by Radicals
The Galois Group of a Polynomial
The Discriminant
Cyclic Extensions
Applications to Polynomial Equations
Reduction Modulo p
Theorem of Reduction Modulo p
Specialization of the Galois Group
Sums and Products of Integers
Norm of the Elements of A
Decomposition Groups
Cyclotomy and Reduction Modulo p
The Chebotarev Theorem
Complements
Zorn's Lemma and Applications
Galois Group of Composite Extensions
Transcendence of e and π
Transcendence of e
Transcendence of π
The Galois Group Over Q of a Polynomial with Integer Coefficients
Symmetric Polynomials
Some Words on Inverse Galois Theory
The Finite Abelian Case
The First Non-abelian Non-trivial Case
The Finite Reductive Case
Some Quotients of G
Review Exercises
Solutions to Exercises
References
Index