Lectures in Abstract Algebra : III. Theory of Fields and Galois Theory / by N. Jacobson.

Format:

Book

Author/Creator:

Jacobson, Nathan, 1910-1999, author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 2197-5612 ; 32.

Graduate Texts in Mathematics, 2197-5612 ; 32

Language:

English

Subjects (All):

Algebra.

Local Subjects:

Algebra.

Physical Description:

1 online resource (XII, 324 pages).

Edition:

First edition 1964.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1964.

System Details:

text file PDF

Summary:

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 second volume, dealing with linear algebra. The level of the present work is roughly the same as that of Volume II. In preparing this book we have had a number of objectives in mind. First and foremost has been that of presenting the basic field theory which is essential for an understanding of modern algebraic number theory, ring theory, and algebraic geometry. The parts of the book concerned with this aspect of the subject are Chapters I, IV, and V dealing respectively with finite dimen- sional field extensions and Galois theory, general structure theory of fields, and valuation theory. Also the results of Chapter IlIon abelian extensions, although of a somewhat specialized nature, are of interest in number theory. A second objective of our ac- count has been to indicate the links between the present theory of fields and the classical problems which led to its development.

Contents:

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

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

11. Artin-Schreier characterization of real closed fields

Suggestions for further reading.

Other Format:

Printed edition:

ISBN:

9781461298724

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