Multi-Valued Fields Book

Ch. 1	Valuation Rings	1
1.1	Valuation Rings and Valuations of Fields	1
1.2	Valuation Rings in Algebraic Extensions	11
1.3	Henselian Valuation Rings	20
1.4	Alebgraic Extensions of Valued Fields	31
1.5	Immediate Extensions	46
1.6	Density	65
1.7	Constructions	71
Ch. 2	Multi-Valued Fields	85
2.1	Prufer Rings	85
2.2	The Zariski Topology and the Restriction Mapping	94
2.3	Affine Families of Valuation Rings	100
2.4	Weakly Boolean Families and Boolean Families of Valuation Rings	111
2.5	Near Boolean Families of Valuation Rings	123
2.6	Independence	130
Ch. 3	Local-Global Properties of Near Boolean Families	141
3.1	Rational Points over Henselian Fields	141
3.2	Arithmetic Local-Global Principle	151
3.3	Equivalent Forms of Property LG[subscript A]	158
3.4	Preservation of Property LG[subscript A] under Algebraic Separable Extensions	167
3.5	Geometric Local-Global Principle	174
3.6	Existence of Families with Property LG[subscript A]	184
Ch. 4	Model-Theoretic Properties of Multi-Valued Fields	195
4.1	Embedding Theorems	195
4.2	The Robinson Theorem	204
4.3	Valued Fields	207
4.4	Finitely Multi-Valued Fields	216
4.5	Multi-Valued Fields with Boolean Families	226
4.6	Multi-Valued Fields with Near Boolean Families	246
	Bibliographical and Historical Remarks	259
	References	263
	Subject Index	269