By Gabriel Daniel Villa Salvador
The fields of algebraic features of 1 variable seem in different parts of arithmetic: complicated research, algebraic geometry, and quantity concept. this article adopts the latter standpoint by way of utilizing an arithmetic-algebraic perspective to the examine of functionality fields as a part of the algebraic thought of numbers, the place a functionality box of 1 variable is the analogue of a finite extension of Q, the sector of rational numbers. the writer doesn't forget about the geometric-analytic points of functionality fields, yet leaves an in-depth exam from this attitude to others.
Key themes and lines:
* comprises an introductory bankruptcy on algebraic and numerical antecedents, together with transcendental extensions of fields, absolute values on Q, and Riemann surfaces
* makes a speciality of the Riemann–Roch theorem, protecting divisors, adeles or repartitions, Weil differentials, classification walls, and more
* contains chapters on extensions, automorphisms and Galois concept, congruence functionality fields, the Riemann speculation, the Riemann–Hurwitz formulation, purposes of functionality fields to cryptography, category box thought, cyclotomic functionality fields, and Drinfeld modules
* Explains either the similarities and primary modifications among functionality fields and quantity fields
* comprises many workouts and examples to reinforce knowing and inspire extra study
The simply must haves are a easy wisdom of box concept, advanced research, and a few commutative algebra. The e-book can function a textual content for a graduate path in quantity thought or a complicated graduate themes path. however, chapters 1-4 can function the bottom of an introductory undergraduate path for arithmetic majors, whereas chapters 5-9 can help a moment path for complicated undergraduates. Researchers attracted to quantity conception, box concept, and their interactions also will locate the paintings a very good reference.