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 Prรผfer 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 Prรผfer 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
CONTENT
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. Prรผfer 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
