## 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. |

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### Contents

III | 1 |

IV | 2 |

V | 14 |

VI | 22 |

VII | 28 |

VIII | 32 |

IX | 70 |

X | 75 |

XXIX | 207 |

XXX | 216 |

XXXI | 221 |

XXXII | 228 |

XXXIII | 235 |

XXXIV | 236 |

XXXV | 239 |

XXXVI | 248 |

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abelian group absolute value additive group algebraic extension algebraically closed Archimedean assume automorphism bijection bounded Cantor set cardinality Cauchy sequence chain closure complete construction contains continuous contradiction converges Corollary countable cyclic deﬁned Deﬁnition dense diﬀerent direct sum discrete element endomorphism Exercise extension ﬁeld F ﬁlterbase ﬁnite ﬁrst ﬁxed follows Galois Galois extension group G Hausdorﬀ hence homeomorphism implies induced inﬁnite integers inversion isomorphic Lemma Let F locally compact maximal metric multiplicative group natural number non-empty non-trivial non-zero normal subgroup obtain open sets order topology order-preserving ordered ﬁeld ordered group p-adic topology polynomial prime number Proof properties Proposition prove Q×pos quotient rational numbers real closed real numbers Section skew ﬁeld square subﬁeld subgroup subset surjective Theorem topological ﬁeld topological group topological ring topological space torsion unique valuation ring vector space