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.Composites of fields
CHAPTER II: GALOIS THEORY OF EQUATIOIVS
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 p-extensions
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 composites offields
CHAPTER V: VALUATION .THEORY
1.Realvaluations
2.Real valuations of the field of rational numbers
3.Real valuations of (x) which are trivial in
4.Completionofafield
5.Some properties of the field of p-adic numbers
6.Hensel'slemma
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
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 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
I0.Generalized Sturm's theorem.Applications
11.Artin-Sehreier characterization of real closed fields
Suggestions for further reading
Index
