代数的数体とその完備化 Algebraic Number Fields and Their Completions (Textbooks in Mathematics)

Childress, Nancy

Chapman & Hall（2017/12発売）

• 製本 Hardcover:ハードカバー版／ページ数 352 p.／サイズ 40 illus.
• 言語 ENG
• 商品コード 9781439852514
• DDC分類 512

基本説明

Integrates local and global theory to reflect a very modern view of algebraic number theory.

Full Description

This book integrates local and global theory to reflect a very modern view of algebraic number theory. This approach is used whenever possible to make the book as accessible as possible to readers with some background in abstract algebra. The author uses contemporary notation and includes numerous examples and end-of-chapter exercises. Suitable for a course on algebraic number theory or as background reading on class field theory, the book covers such topics as localization, ramification theory, norms, Minkowski theory, the unit group, cyclotomic fields, and Dedekind domains.

Contents

Algebraic IntegersOverviewZ-OrdersPrime and Maximal IdealsIntegral Extensions of RingsDedekind DomainsDedekind DomainsAlgebraic Integers in Quadratic FieldsThe Chinese Remainder theoremFractional IdealsThe Ideal Class GroupUnique Factorization of Ideals in A Dedekind DomainLocalizationMultiplicative SubsetsSemilocal RingsDiscrete Valuation RingsRamification TheoryResidue FieldsThe Fundamental IdentityPrime Factorization in Quadratic FieldsRamification, Inertia and SplittingPrime Factorization in Galois ExtensionsThe Decomposition GroupThe Frobenius AutomorphismQuadratic Example Revisitedp-adic NumbersAbsolute valueValuationsTopological Equivalence of Absolute Valuesp-adic Integersp-adic ExpansionsHensel's LemmaLocal Fields and RamificationLocal FieldsAbsolute Values for Extensions of Local FieldsIntegers in Local FieldsRamification for Local FieldsProducing Totally Ramified Extensions of Local FieldsPrime Factorization in Local FieldsPrime Factorization in Global FieldsNormsNorms of ElementsNorms of Fractional IdealsExtensions of NormsCompatibility of Element and Ideal NormsAn Aside on the Class GroupThe Absolute NormConnections between Global and Local ExtensionsDifferent and DiscriminantThe DifferentFinitely Many Primes RamifyThe DiscriminantDiscriminant and RamificationMinkowski TheoryReal and Complex EmbeddingsLatticesFundamental DomainsMinkowski Lattice Point theoremMinkowski Bound and Finiteness of the Class GroupComputation of the Class NumberThe Unit GroupThe Function .IRoots of UnityDirichlet's Unit theoremExamplesCyclotomic FieldsQ(‾P )Subfields of Q(‾P )Prime Decomposition in Q(‾P )Q(‾P' )Decomposition of P in Cyclotomic Fields Q(‾P' )Quadratic ReciprocityKummer's LemmaAbelian Extensions of QKronecker-WeberProblems appear at the end of each chapter.