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Engler A.J., Prestel A. — Valued Fields
Engler A.J., Prestel A. — Valued Fields

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Название: Valued Fields

Авторы: Engler A.J., Prestel A.

Аннотация:

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.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 2005

Количество страниц: 205

Добавлена в каталог: 18.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathbb{P}'$-procyclic      117
Absolute value,      5
Absolute value, Archimedean      6
Absolute value, dependent      6
Absolute value, non-Archimedean      6
Absolute value, p-adic      6
Algebraically maximal      92
Almost valuation      191
Approximation theorem      8 48
Archimedean      6 26 38
Artin — Schreier polynomial      96
Artin — Whaples      8
Baer — Krull Representation      37
Bounded set      189
Brocker — Prestel      164
Canonical henselian valuation      106
Cauchy sequence      9 50
Chevalley      57
Coarsening      42 104
Coarser, valuation ring      104
Cofinal      50
Cofinality      50
Cofinite      174
Comparable, valuation      104
complete      9 50
Completion      12 50
Composition of valuations      45
Cone      see “Positive cone”
Conjugation Theorem      69
Continuity of roots      53
Convergence      9 50
Convex      36
Convex subgroup      26
Convex, $\leq$-convex      36
Convex, convex hull      36
Convex, P-convex      36
Decomposition field      121
Decomposition group      121
Defect      75
Defectless      75
Degree of inseparability      65
Degree of separability      65
Degree valuation      30
Dependence class      42
Dependent absolute values      6
Dependent valuation rings      42
Diagonal quadratic form      163
Dimension inequality      80
Discrete      27
Discrete (of rank 1)      23
Discriminant      66
Divisible hull      78
Equivalent norms      13
Equivalent valuations      29
Euclidean field      94 100
Exact functor      128
Extension, valuation      28
Extension, valuation ring      59
Field of formal Laurent series      24
Field of p-adic numbers      23
Filter      174
Finer, valuation ring      104
Finitely ramified      92
First exact sequence      113 124
First residue form      170
Formal power series      83
Fundamental inequality      72
Gauss extension      33
Gelfand — Mazur      15
Group, $\mathbb{Z}$-group      159
Group, decomposition      121
Group, inertia      124
Group, Prefer      117
Group, procyclic      115
Group, profinite      115
Group, pronilpotent      115
Group, prosolvable      115
Group, ramification      129
Hensel's lemma      20 88
Henselian valuation,      86
Henselian valuation, N-henselian      85 93
Henselian valuation, p-henselian      94
Henselization      121
Immediate extension      61
Independent absolute values      see “Dependent absolute values”
Index of a subgroup      118
Inertia field      124
Inertia group      124
Inverse limit      115
Inverse system      114
Isometric      163
Isotropic      163
Koenigsmann      109 137
Krasner's Lemma      91
Left-exact contravariant functor      128
Lexicographically ordered      27
Lies over, valuation ring      59
Local parameter      see “Uniformizer”
Maximal Galois p-extension      94
Maximal valued field      92
Neutral element      192
Nilpotent element      192
Non-archimedean absolute value      6
Norm      13
Norm, equivalent      13
Order, profinite      116
Ordered abelian group      25
p-adic absolute value      6
p-adic valuation      18
p-adically closed field      157
p-character group      128
p-component      129
p-extensions      94
p-generic polynomial      96
p-henselian      94
p-rank      131
p-Sylow extension      108
p-Sylow subgroup      118
Place      55
Positive cone      36
Prefer completion      117
Prefer group      117
Primitive polynomial      87
Principal ultrafilter      174
Pro-p completion      117
Pro-p group      115
Procyclic group      115
Profinite group      115
Projective limit      see “Inverse limit”
Projective system      see “Inverse system”
Prolongation, valuation      59
Pronilpotent group      115
Prosolvable group      115
Pseudo-Chauchy sequence      82
Pseudo-complete      82
Pseudo-limit      82
Pure      151
Pythagorean      111
Quadratic form      163
Quadratic module      164
Quadratic system of representatives      37
Ramification field      129
Ramification group      129
Ramification index      61
Rank, ordered group      26
Rank, ordered group, rank 1 valuation      26
Rank, valuation      28
Rank, valuation ring      29
Rational rank      78
Real closed field      99
Real closure      100
Real field      36
Relatively complete      52
Residue class field      19 28
Residue degree      61
Residue forms      170
Restriction, valuation      28
Rigid element, T-rigid      39
Ring of p-adic integers      23
Saturatedness, $\aleph_1$-saturatedness      173
Schmidt, F.K.      103
Second exact sequence      113 129
Semiordering      165
Separable polynomial      52
Supernatural number      116
Support, power series      83
Tamely branching at p      137
Topological field,      187
Topological field, V-topology      187
Trivial absolute value      5
Trivial valuation      28
Trivial, valuation ring      29
Ultrafilter      174
Ultrametric inequality      5
Ultraproduct      177
Uniformizer      23
Usual absolute value      6
Valuation ring,      19
Valuation ring, coarser      104
Valuation ring, dependent      42
Valuation ring, extension      59
Valuation ring, finer      104
Valuation,      28
Valuation, almost      191
Valuation, canonical henselian      106
Valuation, coarsening      42
Valuation, comparable      104
Valuation, composition      45
Valuation, equivalent      29
Valuation, extension      28
Valuation, henselian      86
Valuation, p-adic valuation      18
Valuation, prolongation      59
Valuation, restriction      28
Value group      19 28
Weakly isotropic      164
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