Lectures in Abstract Algebra III. Theory of Fields and Galois Theory
The present volume completes the series of texts on algebra which the author began more than ten years ago. The account of field theory and Galois theory which we give here is based on the notions and results of general algebra which appear in our first volume and on the more elementary parts of the...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
1964, 1964
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| Edition: | 1st ed. 1964 |
| Series: | Graduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 4. The general equation of n-th degree
- 5. Equations with rational coefficients and symmetric group as Galois group
- III: Abelian Extensions
- 1. Cyclotomic fields over the rationals
- 2. Characters of finite commutative groups
- 3. Kummer extensions
- 4. Witt vectors
- 5. Abelian p-extensions
- IV: Structure Theory of Fields
- 1. Algebraically closed fields
- 2. Infinite Galois theory
- 3. Transcendency basis
- 4. Lüroth’s theorem
- 5. Linear disjointness and separating transcendency bases
- 6. Derivations
- 7. Derivations, separability and p-independence
- 8. Galois theory for purely inseparable extensions of exponent one
- 9. Higher derivations
- 10. Tensor products of fields
- 11. Free composites of fields
- V: Valuation Theory
- 1. Real valuations
- 2. Real valuations of the field of rational numbers
- 3. Real valuations of ?(x) which are trivial in ?
- 4. Completion of a field
- 5. Some properties of the field of p-adic numbers
- 6. Hensel’s lemma
- 7. Construction of complete fields with given residue fields
- 8. Ordered groups and valuations
- 9. Valuations, valuation rings, and places
- 10. Characterization of real non-archimedean valuations
- 11. Extension of homomorphisms and valuations
- 12. Application of the extension theorem: Hilbert Nullstellensatz
- 13. Application of the extension theorem: integral closure
- 14. Finite dimensional extensions of complete fields
- 15. Extension of real valuations to finite dimensional extension fields
- 16. Ramification index and residue degree
- VI: Artin-Schreier Theory
- 1. Ordered fields and formally real fields
- 2. Real closed fields
- 3. Sturm’s theorem
- 4. Real closure of an ordered field
- 5. Real algebraic numbers
- 6. Positive definite rational functions
- 7. Formalization of Sturm’s theorem. Resultants
- 8. Decision method for an algebraic curve
- 9. Equations with parameters
- 10. Generalized Sturm’s theorem. Applications
- 1. Extension of homomorphisms
- 2. Algebras
- 3. Tensor products of vector spaces
- 4. Tensor product of algebras
- I: Finite Dimensional Extension Fields
- 1. Some vector spaces associated with mappings of fields
- 2. The Jacobson-Bourbaki correspondence
- 3. Dedekind independence theorem for isomorphisms of a field
- 4. Finite groups of automorphisms
- 5. Splitting field of a polynomial
- 6. Multiple roots. Separable polynomials
- 7. The “fundamental theorem” of Galois theory
- 8. Normal extensions. Normal closures
- 9. Structure of algebraic extensions. Separability
- 10. Degrees of separability and inseparability. Structure of normal extensions
- 11. Primitive elements
- 12. Normal bases
- 13 Finite fields
- 14. Regular representation, trace and norm
- 15. Galois cohomology
- 16 Composites of fields
- II: Galois Theory of Equations
- 1. The Galois group of an equation
- 2. Pure equations
- 3. Galois’ criterion for solvability by radicals
- 11. Artin-Schreier characterization of real closed fields
- Suggestions for further reading