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.