The Classical Fields: Structural Features of the Real and Rational NumbersThe classical fields are the real, rational, complex and p-adic numbers. Each of these fields comprises several intimately interwoven algebraical and topological structures. This comprehensive volume analyzes the interaction and interdependencies of these different aspects. The real and rational numbers are examined additionally with respect to their orderings, and these fields are compared to their non-standard counterparts. Typical substructures and quotients, relevant automorphism groups and many counterexamples are described. Also discussed are completion procedures of chains and of ordered and topological groups, with applications to classical fields. The p-adic numbers are placed in the context of general topological fields: absolute values, valuations and the corresponding topologies are studied, and the classification of all locally compact fields and skew fields is presented. Exercises are provided with hints and solutions at the end of the book. An appendix reviews ordinals and cardinals, duality theory of locally compact Abelian groups and various constructions of fields. |
Contents
Real numbers | 1 |
Multiplication and topology of the real numbers | 100 |
Nonstandard numbers | 154 |
Nonstandard rationals | 158 |
A construction of the real numbers | 159 |
Nonstandard reals | 162 |
Ordering and topology | 164 |
η1fields | 166 |
Completion | 235 |
Completion of chains | 236 |
Completion of ordered groups and fields | 239 |
Completion of topological abelian groups | 248 |
Completion of topological rings and fields | 264 |
The padic numbers | 278 |
The field of padic numbers | 279 |
The additive group of padic numbers | 285 |
Continuity and convergence | 170 |
Topology of the real numbers in nonstandard terms | 173 |
Differentiation | 175 |
Planes and fields | 177 |
Rational numbers | 179 |
The multiplication of the rational numbers | 185 |
Ordering and topology of the rational numbers | 193 |
The rational numbers as a field | 207 |
Ordered groups of rational numbers | 216 |
Addition and topologies of the rational numbers | 221 |
Multiplication and topologies of the rational numbers | 228 |
Common terms and phrases
abelian group absolute value additive group algebraically closed assume automorphism bijection bounded Cantor set cardinality Cauchy sequence chain Chapter closure complete concentrated filterbase construction contains continuous contradiction converges Corollary countable cyclic defined definition denote dense direct sum discrete element endomorphism Exercise extension field F finite follows Galois group G Hausdorff hence homeomorphism implies induced infinite integers inversion isomorphic Lemma Let F locally compact Math maximal metric multiplicative group natural number non-empty non-trivial non-zero obtain open sets order-preserving ordered field ordered group p-adic topology polynomial prime number Proof properties Proposition prove quadratic quotient rational numbers real closed real closed field real numbers roots Section skew field square subfield subgroup subset surjective Theorem topological field topological group topological ring topological space torsion unique valuation ring vector space