Galois Theory through Exercises

Description

This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises). In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading. Preface This Book A Few Words on the Subject The Structure of the Book Advice to the Reader Interrelations of Chaps. 1–15 Contents 1 Solving Algebraic Equations Exercises 1 Using Computers 1 2 Field Extensions Exercises 2 3 Polynomials and Irreducibility Exercises 3 Using Computers 3 4 Algebraic Extensions Exercises 4 Using Computers 4 5 Splitting Fields Exercises 5 Using Computers 5 6 Automorphism Groups of Fields Exercises 6 Using Computers 6 7 Normal Extensions Exercises 7 Using Computers 7 8 Separable Extensions Exercises 8 Using Computers 8 9 Galois Extensions Exercises 9 Using Computers 9 10 Cyclotomic Extensions Exercises 10 Using Computers 10 11 Galois Modules Exercises 11 Using Computers 11 12 Solvable Groups Exercises 12 13 Solvability of Equations Exercises 13 14 Geometric Constructions Exercises 14 15 Computing Galois Groups Exercises 15 16 Supplementary Problems 17 Proofs of the Theorems Theorems of Chap. 1 Theorems of Chap. 2 Theorems of Chap. 3 Theorems of Chap. 4 Theorems of Chap. 5 Theorems of Chap. 6 Theorems of Chap. 7 Theorems of Chap. 8 Theorems of Chap. 9 Theorems of Chap. 10 Theorems of Chap. 11 Theorems of Chap. 12 Theorems of Chap. 13 Theorems of Chap. 14 Theorems of Chap. 15 18 Hints and Answers Problems of Chap.1 Problems of Chap.2 Problems of Chap.3 Problems of Chap.4 Problems of Chap.5 Problems of Chap.6 Problems of Chap.7 Problems of Chap.8 Problems of Chap.9 Problems of Chap.10 Problems of Chap.11 Problems of Chap.12 Problems of Chap.13 Problems of Chap.14 Problems of Chap.15 19 Examples and Selected Solutions Problems of Chap.1 Problems of Chap.2 Problems of Chap.3 Problems of Chap.4 Problems of Chap.5 Problems of Chap.6 Problems of Chap.7 Problems of Chap.8 Problems of Chap.9 Problems of Chap.10 Problems of Chap.11 Problems of Chap.12 Problems of Chap.13 Problems of Chap.14 Problems of Chap.15 Appendix: Groups, Rings and Fields Equivalence Relations Groups Rings Polynomial Rings Fields Modules over Rings The Chinese Remainder Theorem Permutations Group Actions on Sets Symmetric Polynomials Transitive Subgroups of Permutation Groups Some Arithmetical Functions Characters and Pairing Zorn's Lemma References List of Notation Index