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＜電子ブック＞
Topics in the Theory of Algebraic Function Fields / by Gabriel Daniel Villa Salvador
(Mathematics: Theory & Applications)

版	1st ed. 2006.
出版者	(Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser)
出版年	2006
本文言語	英語
大きさ	XVI, 652 p : online resource
著者標目	*Villa Salvador, Gabriel Daniel author SpringerLink (Online service)
件　名	LCSH:Number theory LCSH:Functions of complex variables LCSH:Algebraic geometry LCSH:Algebraic fields LCSH:Polynomials LCSH:Mathematical analysis LCSH:Commutative algebra LCSH:Commutative rings FREE:Number Theory FREE:Functions of a Complex Variable FREE:Algebraic Geometry FREE:Field Theory and Polynomials FREE:Analysis FREE:Commutative Rings and Algebras
一般注記	Algebraic and Numerical Antecedents -- Algebraic Function Fields of One Variable -- The Riemann-Roch Theorem -- Examples -- Extensions and Galois Theory -- Congruence Function Fields -- The Riemann Hypothesis -- Constant and Separable Extensions -- The Riemann-Hurwitz Formula -- Cryptography and Function Fields -- to Class Field Theory -- Cyclotomic Function Fields -- Drinfeld Modules -- Automorphisms and Galois Theory The fields of algebraic functions of one variable appear in several areas of mathematics: complex analysis, algebraic geometry, and number theory. This text adopts the latter perspective by applying an arithmetic-algebraic viewpoint to the study of function fields as part of the algebraic theory of numbers, where a function field of one variable is the analogue of a finite extension of Q, the field of rational numbers. The author does not ignore the geometric-analytic aspects of function fields, but leaves an in-depth examination from this perspective to others. Key topics and features: * Contains an introductory chapter on algebraic and numerical antecedents, including transcendental extensions of fields, absolute values on Q, and Riemann surfaces * Focuses on the Riemann–Roch theorem, covering divisors, adeles or repartitions, Weil differentials, class partitions, and more * Includes chapters on extensions, automorphisms and Galois theory, congruence function fields, the Riemann Hypothesis, the Riemann–Hurwitz Formula, applications of function fields to cryptography, class field theory, cyclotomic function fields, and Drinfeld modules * Explains both the similarities and fundamental differences between function fields and number fields * Includes many exercises and examples to enhance understanding and motivate further study The only prerequisites are a basic knowledge of field theory, complex analysis, and some commutative algebra. The book can serve as a text for a graduate course in number theory or an advanced graduate topics course. Alternatively, chapters 1-4 can serve as the base of an introductory undergraduate course for mathematics majors, while chapters 5-9 can support a second course for advanced undergraduates. Researchers interested in number theory, field theory, and their interactions will also find the work an excellent reference HTTP:URL=https://doi.org/10.1007/0-8176-4515-2

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