INTRODUCTION SECTION 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.Normalbases 13.Finitefields 14.Regular representation,trace and norm 15.Galois cohomology 16.Coites of fields
CHAPTER II: GALOIS THEORY OF EUATIOIVS 1.The Galois group of an equation 2.Pureequations 3.Galois' criterion for solvability by radicals 4.The general equation of n-th degree 5.Equations with rational coefficients and symmetric group as Galoisgroup
CHAPTER Ⅲ: ABELIAN EXTENSlONS 1.Cyclotomic fields over the rationals 2.Characters of finite commutatiye groups 3.Kummer extensions 4.Witt rrectors 5.Abelian -xesions
CHAPTER Ⅳ: STRUCTURE THEORY OF FIELDS 1 Algebraically closed fields 2.Infinite Galois theory 3.Transcendency basis 4.Luroth'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 exponert one 9.Higher derivations 10.Tensor products of fields 11.Free coites offields
CHAPTER V: VALUATIN THEORY 1.Realvaluations 2.Real valuations of the field of rational numbers 3.Real valuations of (x) which are trivial in 4.Comlioofafield 5.Some properties of the field of p-adi ubers 6.Hensel'slemma 7.Construction of complete fields with given residue fields 8.Ordered groups an-vutons 9.Valuations, valuation rings, and places 10.Characterization of real non-archimedean valuations 11.Extension of homomorphisms an vutons 12.Application of the extension theorem: Hilbert Nullstellensatz 13.Application of the extension theorem: integral closure SECTION 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: 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 algebrai ubers 6.Positive definite rational functions 7.Formalization of Sturm's theorem.Resultants 8.Decision method for an algebraic curve 9.Equations with parameters I0.Generalized Sturm's theorem.Applications 11.Artin-Sehreier characterization of real closed fields