Algebraic numbers and algebraic functions
著者
書誌事項
Algebraic numbers and algebraic functions
（Chapman and Hall mathematics series）
Chapman & Hall, 1991
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注記
Bibliography: p. [183]-185
Includes Index
内容説明・目次
内容説明
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.
目次
- Part 1 Fields with valuations: absolute values
- the topology defined by an absolute value
- complete fields
- valuations, valuation rings and places
- the representation by power series
- ordered groups
- general valuations. Part 2 Extensions: generalities on extensions
- extensions of complete fields
- extensions of incomplete fields
- Dedekind domains and the string approximation theorem
- extensions of Dedekind domains
- different and discriminant. Part 3 Global fields: algebraic number fields
- the product formula
- the unit theorem
- the class number. Part 4 Function fields: divisors on a function field
- principal divisors and the divisor class group
- Riemann's theorem and the speciality index
- the genus
- derivations and differentials
- the Riemann-Roch theorem and its consequences
- elliptic function fields
- Abelian integrals and the Abel-Jacobi theorem. Part 5 Algebraic function fields in two variables: valuations on function fields of two variables.
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