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 Riemanna "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
Riemanna "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
mathematicsmajors, 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.
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