Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The later chapters probe the analogy between global function fields and algebraic number fields.
Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.
From the reviews:
MATHEMATICAL REVIEWS
Both in the large (choice and arrangement of the material) and in the details (accuracy and completeness of proofs, quality of explanations and motivating remarks), the author did a marvelous job. His parallel treatment of topics...for both number and function fields demonstrates the strong interaction between the respective arithmetics, and allows for motivation on either side.
Bulletin of the AMS
... Which brings us to the book by Michael Rosen. In it, one has an excellent (and, to the author's knowledge, unique) introduction to the global theory of function fields covering both the classical theory of Artin, Hasse, Weil and presenting an introduction to Drinfeld modules (in particular, the Carlitz module and its exponential). So the reader will find the basic material on function fields and their history (i.e., Weil differentials, the Riemann-Roch Theorem etc.) leading up to Bombieri's proof of the Riemann hypothesis first established by Weil. In addition one finds chapters on Artin's primitive root Conjecture for function fields, Brumer-Stark theory, the ABC Conjecture, results on class numbers and so on. Each chapter contains a list of illuminating exercises. Rosen's book is perfect for graduate students, as well as other mathematicians, fascinated by the amazing similarities between number fields and function fields.
David Goss (Ohio State University)