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Multi-Valued Fields / by Yuri L. Ershov
(Siberian School of Algebra and Logic)

1st ed. 2001.
出版者 New York, NY : Springer US : Imprint: Springer
出版年 2001
本文言語 英語
大きさ X, 270 p : online resource
著者標目 *Ershov, Yuri L author
SpringerLink (Online service)
件 名 LCSH:Algebra
LCSH:Algebraic fields
LCSH:Polynomials
LCSH:Commutative algebra
LCSH:Commutative rings
LCSH:Mathematical logic
FREE:Algebra
FREE:Field Theory and Polynomials
FREE:Commutative Rings and Algebras
FREE:Mathematical Logic and Foundations
一般注記 1. Valuation Rings -- 1.1. Valuation Rings and Valuations of Fields -- 1.2. Valuation Rings in Algebraic Extensions -- 1.3. Henselian Valuation Rings -- 1.4. Alebgraic Extensions of Valued Fields -- 1.5. Immediate Extensions -- 1.6. Density -- 1.7. Constructions -- 2. Multi-Valued Fields -- 2.1. Prüfer Rings -- 2.2. The Zariski Topology and the Restriction Mapping -- 2.3. Affine Families of Valuation Rings -- 2.4. Weakly Boolean Families and Boolean Families of Valuation Rings -- 2.5. Near Boolean Families of Valuation Rings -- 2.6. Independence -- 3. Local-Global Properties of Near Boolean Families -- 3.1. Rational Points over Henselian Fields -- 3.2. Arithmetic Local-Global Principle -- 3.3. Equivalent Forms of Property LGA -- 3.4. Preservation of Property LGA under Algebraic Separable Extensions -- 3.5. Geometric Local-Global Principle -- 3.6. Existence of Families with Property LGA -- 4. Model-Theoretic Properties of Multi-Valued Fields -- 4.1. Embedding Theorems -- 4.2. The Robinson Theorem -- 4.3. Valued Fields -- 4.4. Finitely Multi-Valued Fields -- 4.5. Multi-Valued Fields with Boolean Families -- 4.6. Multi-Valued Fields with Near Boolean Families -- Bibliographical and Historical Remarks -- References
For more than 30 years, the author has studied the model-theoretic aspects of the theory of valued fields and multi-valued fields. Many of the key results included in this book were obtained by the author whilst preparing the manuscript. Thus the unique overview of the theory, as developed in the book, has been previously unavailable. The book deals with the theory of valued fields and mutli-valued fields. The theory of Prüfer rings is discussed from the `geometric' point of view. The author shows that by introducing the Zariski topology on families of valuation rings, it is possible to distinguish two important subfamilies of Prüfer rings that correspond to Boolean and near Boolean families of valuation rings. Also, algebraic and model-theoretic properties of multi-valued fields with near Boolean families of valuation rings satisfying the local-global principle are studied. It is important that this principle is elementary, i.e., it can be expressed in the language of predicate calculus. The most important results obtained in the book include a criterion for the elementarity of an embedding of a multi-valued field and a criterion for the elementary equivalence for multi-valued fields from the class defined by the additional natural elementary conditions (absolute unramification, maximality and almost continuity of local elementary properties). The book concludes with a brief chapter discussing the bibliographic references available on the material presented, and a short history of the major developments within the field
HTTP:URL=https://doi.org/10.1007/978-1-4615-1307-0
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Springer eBooks 9781461513070
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データ種別 電子ブック
分 類 LCC:QA150-272
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書誌ID 4000106330
ISBN 9781461513070

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