Multi-Valued Fields

• 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.

From the reviews: "The book is devoted to the investigation of fields with distinguished families of valuation rings. The author has been studying the model-theoretic aspects of the theory of valued fields and multi-valued fields for more than 30 years. Many of the key results included in the book are published for the first time. So, the unique overview of the theory presented in the book has been unavailable previously." (German Pestov, Zentralblatt MATH, Vol. 1073 (24), 2005)