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Bachman G. — Introduction to p-Adic Numbers and Valuation Theory
Bachman G. — Introduction to p-Adic Numbers and Valuation Theory



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Название: Introduction to p-Adic Numbers and Valuation Theory

Автор: Bachman G.

Аннотация:

The book is meant to serve as an introduction to valuation theory. The first two chapters have been written mainly for advanced undergraduate students and first year graduate students.The amount of algebra required is quite small, and the algebraic results needed for these two chapters are included in the first four sections of the appendix. It is hoped that in this fashion these two chapters will be reasonably self-contained and available to as wide an audience as possible.
The remaining three chapters definitely demand more mathematical maturity on the part of the reader. At least a first course in modern algebra would be required to read parts of them. (From author's preface)


Язык: en

Рубрика: Математика/Теория чисел/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1964

Количество страниц: 173

Добавлена в каталог: 20.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute convergence      98
Adeles      167
Algebraic closure      166
Algebraic integer      89
Analyticity      114
Approximation theorem      21
Archimedian ordered group      78
Associated place      69
Associated valuation      74
Associated valuation ring      68 73
Banach algebra      111
Banach space      94
Binomial series      51
Bounded linear functional      100
Canonical homomorphism      160 163
Cauchy sequence      16 94
Characteristic      164
Chinese remainder theorem      22 136
Commutative normed algebra      111
Complete field      24
Complete residue system      156
Completion of a field      26—33
Congruence      26
Conjugate embeddings      130
Convergence, absolute      98
Convergence, p-adic valuation      4
Convergence, rank one valuation      24
Convex functional      103
Degree of separability      166
Degree, extension      165
Discrete valuation      143
Discriminant      142
Divisibility      164
Domain of convergence      43
Double coset      161
Eisenstein criteria      151
Equivalent norms      94
Equivalent places      70
Equivalent valuations      16
Euclidean Domain      165
Euler $\varphi$-function      156
Euler theorem      156
Exponential series      46 47
Extension of a valuation, archimedian      126—128
Extension of a valuation, non-archimedian      117—126
Extension, algebraic      165
Extension, field      165
Extension, finite      165
Extension, normal      165
Extension, separable      165
Extension, theorem      83—88
Extension, transcendental      165
Fermat's theorem      156
Fully ramified extension      152
Gelfand theorem      116
Global degree      132—133
Global norm      136
Global trace      136
Greatest common divisor (g.c.d.)      155—156
Group of units      23 67
Group, Galois      166
Group, ordered      70
Hahn — Banach theorem      103 106 107
Ideal      163
Ideles      168
Identity element      111
Induced valuation      130
Inductively ordered      155
Integral closure      89
Integral element      88
Irreducible element      164
Isolated subgroup      77
Isometry      26
Kernel      161
Lagrange's theorem      159
Least common multiple      156
Legendre symbol      157
Linear functional      99 100
Linear functional, norm of      101
Liouville's theorem      114
Local degree      133
Local norm      136
Local trace      136
Logarithmic series      48 49
Metric      3 4 15 93
Metric spaces      3 4 15 93
Neighborhood (spherical)      15
Newton's method      52—57
Non-units      65
Norm      92 166
Normalized p-adic valuation      23
Normed algebra      111
Normed linear space      92
Null sequence      16
Order isomorphism      78
Ordered group      70 77
Ordinal      45
p-Adic integer      37
p-adic numbers      33 34
p-adic numbers (canonical expansion of)      35
p-adic valuation      2
Place      67—70
Power series in $Q_p$      43
Prime      164
Prime field      164
Principal ideal domain      165
Quadratic non-residue      157
Quadratic reciprocity law      158
Quadratic residue      157
Radius of convergence (power series)      43 98
Ramification index      121
Rank of a valuation      77
Rank of an ordered group      77
Regular point      113
Relatively prime      156
Residue, class degree      121
Residue, class field      9 67
Residue, classes      156
Roots of unity in $Q_p$      61
Semigroup      158
Spectrum      113
Splitting field      166
Subgroup      158
Subgroup, normal      160
Sublinear functional      103
Symmetric convex functional      107
Trace      167
Trivial valuation ring      65
Ultra-metric inequality      3 4
Unique factorization domain      165
UNIT      162
Units, Group of      23 67
Valuation of algebraic number field      137
Valuation of rank one      5
Valuation, associated      74
Valuation, discrete      143
Valuation, equivalent      16 75
Valuation, general      72
Valuation, induced by an embedding      130
Valuation, non-archimedian      5 72
Valuation, normalized p-adic      23
Valuation, p-adic      2 5
Valuation, ring      9 65
Valuation, trivial      9
Valuation, vector      168
Zorn's lemma      155
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