| introduction 1. extension of homomorphisms 2. algebras 3. tensor products of vector spaces 4. tensor product of algebras chapter 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 chapter ii: oalois 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 chapter 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 chapter iv: structure theory of fields 1. algebraically closed fields 2. infinite galois theory 3. transcendency basis 4. lfiroth'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 chapter 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 chapter vi:arti-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 index |
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