Cohn P.M. — Algebraic numbers and algebraic functions :: Электронная библиотека попечительского совета мехмата МГУ

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Cohn P.M. — Algebraic numbers and algebraic functions

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Название: Algebraic numbers and algebraic functions

Автор: Cohn P.M.

Аннотация:

This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.

Язык:

Рубрика: Математика/Теория чисел/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 202

Добавлена в каталог: 28.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

1-unit      26
Abel — Jacobi theorem      155 172
Abel's theorem      163
Absolute norm, unit      101ff.
Absolute residue degree      110
Absolute value      1
Addition theorem      157 162
Adele      126
Algebraic function field      109
Algebraic integer      84
Algebraic number      83ff.
Analytic isomorphism      7
Anharmonic case (elliptic function field)      165
Approximation theorem      8
Archimedean      3
Archimedean ordering      33
Artin, E. (1898—1962)      10 15 83
Associated elements      21
atom      71
based      114
Bloch, A. (1893—1948)      163
Branchpoint      111
Canonical divisor class      139
Cantor, G. (1845—1918)      83
Casorati — Weierstrass theorem      163
Cauchy sequence      12
Cauchy's theorem      140 168
Chain rule      136
Chevalley, C. (1909—1984)      47 105
Chinese remainder theorem      8
Class number      106
Complement      72
Complete, completion      12
Completion by cuts      32
Composite of fields      55
Conorm map      70
Constants (field of)      98
Convergent      11
Convex subgroup      31
Coprime      115
Critical order      139
Cubical norm      14
Decomposed prime      88
Dedekind cut      32
Dedekind discriminant theorem      79
Dedekind domain      64
Dedekind, J.W.R. (1831—1916)      32 91
Deficiency      129
Degree      85 110 115 123
Denominator      115 118
Derivation      135
Different      74
Differential      137
Differential, first kind      145
Differential, second, third kind      162
Direct product      55
Dirichlet, P.G.L. (1805—1859)      105
Discrete ordered group      34
Discrete subgroup of $\mathbb{R}^n$       103
Discrete valuation      19
Discriminant      76
divides      67 115
Divisor (group)      61 114
Divisor class group      69
Divisor of poles, zeros      118
Domination      20
Dual basis      73
Einseinheit      26
Elliptic function field      130
Elliptic integral      160
Elliptic modular function      164
Equivalent absolute values      7
Equivalent valuations      21
Euclidean algorithm      89
Euler, L. (1707—1783)      90
Fermat's last theorem      134
Fermat, P. de (1601—1665)      90
Field composite      55
Field of constants      98
Fractional ideal      61
Function field      109
Fundamental domain      161
Gap      148
Gauss's lemma      85
Gaussian extension      47
Gaussian integer      84
Gelfand — Mazur theorem      16
General valuation      19
Genus      125 129 146ff.
Global field      95 -97
Golod, E.S.      108
Group algebra      41
Haar integral      26
Hasse, H. (1898—1980)      105
Hausdorff separation axiom      10
Hensel's lemma      47 51
Hensel, K. (1861—1941)      2
Hermite, C. (1822—1901)      84
Hilbert's theory of ramification      81
Hilbert, D. (1862—1943)      108
Hurwitz' formula      147
Hurwitz, A. (1859—1919)      154 167
Hyperelliptic field      149
Ideal class group      106
Inert prime      88
Inertia subgroup      54
Integral at p      59
Integral closure, element      48
Integral divisor      115
Integral ideal      61
Invertible ideal      61
Irreducible element      71
Isolated divisor      124
Isomorphic places      24
Jacobi's theorem      174
Jacobian variety      123 171
Koebe, P. (1882—1945)      168
Krull valuation      19
Krull, W. (1899—1971)      18

Lagrange interpolation formula      75
Laurent series      30
Lemniscate      165
Lexicographic ordering      34ff.
Lindemann, C.L.F. v. (1852—1939)      84
Linear series      124 128 158
Liouville's theorem      114
Liouville, J. (1809—1882)      84
Local field      99
Local ring      38
localization      38
Lower segment      32
Lueroth's theorem      148
Majorize      32
Manifold      131
Mazur, S. (1905—1981)      16
Metric space      6
Minkowski constant      108
Minkowski's theorem      100
Minkowski, H. (1864—1909)      105
Mittag — Leffler's theorem      112
Multiplicative group of a field      89
Noether's theorem      148
Noether, E. (1882—1935)      58 64ff.
Noether, M. (1844—1921)      150
Noetherian ring      64
Non-archimedean      3
Norm      5 76
Norm map      70
Normalized valuation      19
Normed vector space      13
Numerator      115 118
Order      3
Order of a parallelotope      95
Order-homomorphism      31
Ordered field      42
Ordered group      31
Ostrowski, A.A893—1986)      15 17
Over      22 24
p-adic value      2
Parallelotope      93
Pell's equation      106
Period (lattice)      155 161
Period matrix      168
Picard variety      171
Picard's theorem      163ff.
PID = principal ideal domain      71 87
Place      24 58 92 110
Pole      3 111 138
Positive      31
Prime divisor      58 61 114
Prime element      22 71
Primitive divisor      124
Primitive polynomial      84
Principal divisor      118
Principal divisor class      145
Principal valuation (ring)      19 21
Principle of domination      20
Product formula      92
Puiseux's theorem      112
Quadratic extension      85
Ramification divisor      113
Ramification index      43
Ramification point      111
Ramified      44 88
Rank      19 32
Rational rank      37 53
Real-valued      19
Regular point      111
Residue      140
Residue class field      20
Residue degree      43
Riemann matrix (pair)      170
Riemann sphere      24
Riemann surface      111 131
Riemann — Roch formula, theorem      112 144
Riemann's theorem      125
Roch, G. (1839—1866)      144
Schmidt, F.K. (1901—1977)      99
Segment, upper, lower      32
Separating element      99
Series, linear      124 128 158
Serre duality theorem      143
Shafarevich, I.R. (1923-)      108
Specially index)      126 145
Steinitz, E. (1871—1928)      35
Stokes' theorem      168
Strong approximation property, theorem      59ff.
Subordinate valuation      38
Supernatural number      35 180
Support      42
Symmetry      158
Tamely ramified      81 112
Tensor product      54
Transcendence degree      51
Transcendental      83
Translation      159
Translation invariance      6
Triangle inequality      1 6
Trijection      25
Trivial      19
UFD = unique factorization domain      71
Ultrametric inequality      3
Uniformizer      22
Unit divisor      114
Unit theorem      102
Upper segment      32
Valuated field      19
Valuation      19
Valuation ring, integers      20
Value (at a place)      24 111
Value group      19ff.
Volume      94
Weierstrass gap theorem      150
Weierstrass normal form      156
Weierstrass point      150
Weierstrass, K.T.W. (1815—1897)      163
Whaples, G. (1914—1981)      83
Zero      3 111 138

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