The purpose of this book is twofold. First, it is written to be a textbook for a graduate level course on Galois theory or field theory. Second, it is designed to be a reference for researchers who need to know field theory. The book is written at the level of students who have familiarity with the basic concepts of group, ring, vector space theory, including the Sylow theorems, factorization in polynomial rings, and theorems about bases of vector spaces. This book has a large number of examples and exercises, a large number of topics covered, and complete proofs given for the stated results. To help readers grasp field.
Author(s): Patrick Morandi
Series: Graduate Texts in Mathematics volume 167
Edition: 1
Publisher: Springer
Year: 1996
Language: English
Pages: 299
Preface
Notes to the Reader
Contents
List of Symbols
Chapter 1. Galois Theory
1 - Field Extensions
2 - Automorphisms
3 - Normal Extensions
4 - Separable and Inseparable Extensions
5 - The Fundamental Theorem of Galois Theory
Chapter 2. Some Galois Extensions
6 - Finite Fields
7 - Cyclotomic Extensions
8 - Norms and Traces
9 - Cyclic Extensions
10 - Hilbert Theorem 90 and Group Cohomology
11 - Kummer Extensions
Chapter 3. Applications of Galois Theory
12 - Discriminants
13 - Polynomials of Degree 3 and 4
14 - The Transcendence of π and e
15 - Ruler and Compass Constructions
16 - Solvability by Radicals
Chapter 4. Infinite Algebraic Extensions
17 - Infinite Galois Extensions
18 - Some Infinite Galois Extensions
Chapter 5. Transcendental Extensions
19 - Transcendence Bases
20 - Linear Disjointness
21 - Algebraic Varieties
22 - Algebraic Function Fields
23 - Derivations and Differentials
Appendix A. Ring Theory
1 - Prime and Maximal Ideals
2 - Unique Factorization Domains
3 - Polynomials over a Field
4 - Factorization in Polynomial Rings
5 - Irreducibility Tests
Appendix B. Set Theory
1 - Zorn’s Lemma
2 - Cardinality and Cardinal Arithmetic
Appendix C. Group Theory
1 - Fundamentals of Finite Groups
2 - The Sylow Theorems
3 - Solvable Groups
4 - Profinite Groups
Appendix D. Vector Spaces
1 - Bases and Dimension
2 - Linear Transformations
3 - Systems of Linear Equations and Determinants
4 - Tensor Products
Appendix E. Topology
1 - Topological Spaces
2 - Topological Properties
References
Index
