Topics in the Theory of Algebraic Function Fields [electronic resource] / by Gabriel Daniel Villa Salvador.
Contributor(s): SpringerLink (Online service)Material type: TextSeries: Mathematics: Theory & Applications: Publisher: Boston, MA : Birkhäuser Boston, 2006Description: XVI, 652 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780817645151Subject(s): Mathematics | Algebraic geometry | Commutative algebra | Commutative rings | Algebra | Field theory (Physics) | Mathematical analysis | Analysis (Mathematics) | Functions of complex variables | Number theory | Mathematics | Number Theory | Functions of a Complex Variable | Algebraic Geometry | Field Theory and Polynomials | Analysis | Commutative Rings and AlgebrasAdditional physical formats: Printed edition:: No titleDDC classification: 512.7 LOC classification: QA241-247.5Online resources: Click here to access online
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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.