Valued FieldsSpringer Science & Business Media, 28.12.2005 - 208 Seiten Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization. In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge acquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -for instance to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values alone. |
Inhalt
| 2 | |
2 | 25 |
3 | 62 |
Henselian Fields | 85 |
4 | 104 |
5 | 112 |
6 | 153 |
Ultraproducts of Valued Fields | 173 |
Classification of VTopologies | 187 |
Standard Notations | 201 |
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Häufige Begriffe und Wortgruppen
abelian group absolute Galois group algebraic closure algebraic extension assume assumption called Cauchy sequence char Clearly closed residue class contains contradiction convex subgroup Corollary define degree denote element euclidean exists extends uniquely extension L/K fact finite extension fixed field follows G₁ Galois extension Hence Hensel's Lemma henselian field henselian valuation ring henselization homomorphism immediate extension implies induced irreducible isomorphism K₁ M₁ maximal ideal minimal polynomial Moreover nilpotent non-archimedean absolute value O₁ O1 and O2 obtain ordered abelian group p-adically closed fields p-henselian p-Sylow polynomial ƒ prime pro-p group profinite group Proposition prove rank real closed residue class field satisfies Sect semiordering separable polynomial separably closed surjective T₁ tamely branching topology trivial ultraproduct unique extension value group value-preserving valued field zero