Author | Jacobson, Nathan. author |
---|---|
Title | Lectures in Abstract Algebra [electronic resource] : III. Theory of Fields and Galois Theory / by Nathan Jacobson |
Imprint | New York, NY : Springer New York, 1964 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-9872-4 |
Descript | XII, 324 p. online resource |
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 โfundamental theoremโ 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. Galoisโ 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. Lรผrothโs 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. Henselโs 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. Sturmโs theorem -- 4. Real closure of an ordered field -- 5. Real algebraic numbers -- 6. Positive definite rational functions -- 7. Formalization of Sturmโs theorem. Resultants -- 8. Decision method for an algebraic curve -- 9. Equations with parameters -- 10. Generalized Sturmโs theorem. Applications -- 11. Artin-Schreier characterization of real closed fields -- Suggestions for further reading