دانلود کتاب Galois Theory, Second Edition

Praise for the First Edition". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!"—Monatshefte fur MathematikGalois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel’s theory of Abelian equations, casus irreducibili, and the Galois theory of origami.In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including:The contributions of Lagrange, Galois, and KroneckerHow to compute Galois groupsGalois's results about irreducible polynomials of prime or prime-squared degreeAbel's theorem about geometric constructions on the lemniscatesGalois groups of quartic polynomials in all characteristicsThroughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study.Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.ContentsPreface to the First Edition xviiPreface to the Second Edition xxiNotation xxiii1 Basic Notation xxiii2 Chapter-by-Chapter Notation xxvPART I POLYNOMIALS1 Cubic Equations 31.1 Cardan's Formulas 41.2 Permutations of the Roots 101.3 Cubic Equations over the Real Numbers 152 Symmetric Polynomials 252.1 Polynomials of Several Variables 252.2 Symmetric Polynomials 302.3 Computing with Symmetric Polynomials (Optional) 422.4 The Discriminant 463 Roots of Polynomials 553.1 The Existence of Roots 553.2 The Fundamental Theorem of Algebra 62PART II FIELDS4 Extension Fields 734.1 Elements of Extension Fields 734.2 Irreducible Polynomials 814.3 The Degree of an Extension 894.4 Algebraic Extensions 955 Normal and Separable Extensions 1015.1 Splitting Fields 1015.2 Normal Extensions 1075.3 Separable Extensions 1095.4 Theorem of the Primitive Element 1196 The Galois Group 1256.1 Definition of the Galois Group 1256.2 Galois Groups of Splitting Fields 1306.3 Permutations of the Roots 1326.4 Examples of Galois Groups 1366.5 Abelian Equations (Optional) 1437 The Galois Correspondence 1477.1 Galois Extensions 1477.2 Normal Subgroups and Normal Extensions 1547.3 The Fundamental Theorem of Galois Theory 1617.4 First Applications 1677.5 Automorphisms and Geometry (Optional) 173PART III APPLICATIONS8 Solvability by Radicals 1918.1 Solvable Groups 1918.2 Radical and Solvable Extensions 1968.3 Solvable Extensions and Solvable Groups 2018.4 Simple Groups 2108.5 Solving Polynomials by Radicals 2158.6 The Casus Irreducbilis (Optional) 2209 Cyclotomic Extensions 2299.1 Cyclotomic Polynomials 2299.2 Gauss and Roots of Unity (Optional) 23810 Geometric Constructions 25510.1 Constructible Numbers 25510.2 Regular Polygons and Roots of Unity 27010.3 Origami (Optional) 27411 Finite Fields 29111.1 The Structure of Finite Fields 29111.2 Irreducible Polynomials over Finite Fields (Optional) 301PART IV FURTHER TOPICS12 Lagrange, Galois, and Kronecker 31512.1 Lagrange 31512.2 Galois 33412.3 Kronecker 34713 Computing Galois Groups 35713.1 Quartic Polynomials 35713.2 Quintic Polynomials 36813.3 Resolvents 38613.4 Other Methods 40014 Solvable Permutation Groups 41314.1 Polynomials of Prime Degree 41314.2 Imprimitive Polynomials of Prime-Squared Degree 41914.3 Primitive Permutation Groups 42914.4 Primitive Polynomials of Prime-Squared Degree 44415 The Lemniscate 46315.1 Division Points and Arc Length 46415.2 The Lemniscatic Function 47015.3 The Complex Lemniscatic Function 48215.4 Complex Multiplication 48915.5 Abel's Theorem 504A Abstract Algebra 515A.1 Basic Algebra 515A.2 Complex Numbers 524A.3 Polynomials with Rational Coefficients 528A.4 Group Actions 530A.5 More Algebra 532Index 557