Topics in Field Theory (Notas De Matematica, 124)
By Gregory Karpilovsky
* Publisher: North-Holland
* Number Of Pages: 558
* Publication Date: 1989-03
* ISBN-10 / ASIN: 0444872973
* ISBN-13 / EAN: 9780444872975
Contents
PREFACE
CHAPTER 1. ALGEBRAIC PRELIMINARIES
1. Notation and terminology
2. Localization
3. Integral extensions
4. Polynomial rings
5. Unique factorization domains
6. Dedekind domains
CHAPTER 2. SEPARABLE ALGEBRAIC EXTENSIONS
1. Algebraic closure, splitting fields and normal extensions
2. Separable algebraic extensions: definitions and elementary properties
3. Separability, linear disjointness and tensor products
4. Norms, traces and discriminants of separable field extensions
CHAPTER 3. TRANSCENDENTAL EXTENSIONS
1. Abstract dependence relations
2. Transcendency bases
3. Simple transcendental extensions
4 . Separable extensions
5. Weil’s order of inseparability
6. Separability and preservation of pindependence
7. Perfect ground fields
8. Criteria for separating transcendency bases
9. Separable generation of intermediate field extensions
10. The Steinitz field tower
11. Nonseparably generated fields over maximal perfect subfields
12. Relatively separated extensions
13. Reliability and relative separability
CHAPTER 4. DERIVATIONS
1. Definitions and elementary properties
2. Extensions of derivations
3. Derivations, separability and pindependence
4. Restricted subspace of DerE
193
193
203
210
213
CHAPTER 5. PURELY INSEPARABLE EXTENSIONS 219
1. Preparatory results for splitting theory 219
2. Splitting theory 234
3. Chains of splitting fields and complexity 244
4. Modular extensions 252
A. Introduction and preliminary results 252
B. Pure independence, basic subfields and tensor products of simple extensions 262
C. Ulm invariants of modular extensions 269
D. Ulm invariants and group algebras 276
E. Modular closure and modularly perfect fields 290
CHAPTER 6. GALOIS THEORY
1. Topological prerequisites
2. Profinite groups
3. Galois extensions
4. Finite fields, roots of unity and cyclotomic extensions
5. Finite Galois theory
6. Infinite Galois theory
7. Realizing finite groups as Galois groups
8. Degrees of sums in a separable field extension
9. Galois cohomology
CHAPTER 7. ABELIAN EXTENSIONS
1. Witt vectors
2. Cyclic extensions
3. Abelian pextensions over fields of characteristic p
4. Kummer theory
5. Character groups of infinite abelian extensions
CHAPTER 8. RADICAL EXTENSIONS
1. Irreducibility of binomials and applications
2. Solvability of Galois groups of radical extensions
3. Abelian binomials
4. Normal binomials
5. Some additional results
6. Cogalois extensions
7. A Galois correspondence for radical extensions
8. Duality of lattices for Gal(E/F) and Cog(E/F)
9. The lattice of intermediate fields of radical extensions
Bibliography
Notation
Index
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