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ThepresentVrolumecompletestheseriesoftextsonalgebrawhichtheauthorbeganmorethantenyearsago.TheaccountoffieldtheoryandGaloistheorywhichwegirrehereisbasedonthenotionsandresultsofgeneralalgebrawhichappearinourfirstvolumeandonthemoreelementarypartsofthesecondvolume，dealingwithlinearalgebra.Thelevelofthepresentwork.isroughlythesameasthatofVolumeII.
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INTRODUCTION
SECTION
1.Extensionofhomomorphisms
2.Algebras
3.Tensorproductsofvectorspaces
4.Tensorproductofalgebras
CHAPTERI:FINITEDIMENSIONALEXTENSIONFIELDS
1Somevectorspacesassociatedwithmappingsoffields
2.TheJacobson-Bourbakicorrespondence
3.Dedekindindependencetheoremforisomorphismsofafield
4.Finitegroupsofautomorphisms.
5.Splittingfieldofapolynomial
6.Multipleroots.Separablepolynomials
7.The"fundamentaltheorem"ofGaloistheory
8.Normalextensions.Normalclosures
9.Structureofalgebraicextensions.Separability
10.Degreesofseparabilityandinseparability.Structureofnormalextensions
11.Primitiveelements
12.Normalbases
13.Finitefields
14.Regularrepresentation,traceandnorm
15.Galoiscohomology
16.Compositesoffields
CHAPTERII:GALOISTHEORYOFEQUATIOIVS
1.TheGaloisgroupofanequation
2.Pureequations
3.Galois''criterionforsolvabilitybyradicals
4.Thegeneralequationofn-thdegree
5.EquationswithrationalcoefficientsandsymmetricgroupasGaloisgroup
CHAPTERⅢ:ABELIANEXTENSlONS
1.Cyclotomicfieldsovertherationals
2.Charactersoffinitecommutatiyegroups
3.Kummerextensions
4.Wittrrectors
5.Abelianp-extensions
CHAPTERⅣ:STRUCTURETHEORYOFFIELDS
1Algebraicallyclosedfields
2.InfiniteGaloistheory
3.Transcendencybasis
4.Luroth''stheorem.
5.Lineardisjointnessandseparatingtranscendencybases
6.Derivations
7.Derivations,separabilityandp-independence
8.Galoistheoryforpurelyinseparableextensionsofexponertone
9.Higherderivations
10.Tensorproductsoffields
11.Freecompositesoffields
CHAPTERV:VALUATION.THEORY
1.Realvaluations
2.Realvaluationsofthefieldofrationalnumbers
3.Realvaluationsof（x）whicharetrivialin
4.Completionofafield
5.Somepropertiesofthefieldofp-adicnumbers
6.Hensel''slemma
7.Constructionofcompletefieldswithgivenresiduefields
8.Orderedgroupsand-valuations
9.Valuations,valuationrings,andplaces
10.Characterizationofrealnon-archimedeanvaluations
11.Extensionofhomomorphismsandvaluations
12.Applicationoftheextensiontheorem:HilbertNullstellensatz
13.Applicationoftheextensiontheorem:integralclosure
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CHAPTERVI:ARTIN-SCHREIERTHEORY
Index
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