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hernandez d laszlo y introduction galois theory 2024

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Textbook in PDF format 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

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hernandez d laszlo y introduction galois theory 2024

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