抽象代数讲义(第3卷)
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作者(德)雅格布斯
出版社世界图书出版公司
ISBN9787510061530
出版时间2013-10
装帧其他
开本其他
定价69元
货号2650860
上书时间2024-10-14
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导语摘要
《抽象代数讲义》是一套久负盛名的三卷集教材，是作者雅格布斯根据他在霍普金斯大学和耶鲁大学讲课时的讲义编写而成的，后又成为作者《基本代数学》一书的蓝本。《抽象代数讲义（第3卷）》介绍域理论和伽罗瓦理论，讨论了域的代数结构和域的赋值理论。本书为全英文版本。
目录
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
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