Topics in the Theory of Algebraic Function Fields [Hardcover]

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. The examination explains both the similarities and fundamental differences between function fields and number fields, including 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.

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.

From the reviews:

Gabriel Daniel Villa Salvador's book Topics in the Theory of Algebraic Function Fields was recently translated into English and published by Birkh?user (the original version was published in Mexico in 2003). In the book, Salvador covers a large amount of material at a wide range of levels of difficulty, and as such his book could be used in a wide variety of settings.... The entire book is very well written and Villa Salvador includes