Titlebook: Number Theory in Function Fields; Michael Rosen Textbook 2002 Springer Science+Business Media New York 2002 Algebraic Function Fields.Alge

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Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem,be presented in a geometric fashion. Function fields correspond to algebraic curves and finite extensions of function fields correspond to ramified covers of curves. In this chapter, however, we will continue to use a more arithmetic point of view which emphasizes the analogy of function fields with

FLAX

Constant Field Extensions,ith constant field .. For every field extension . of . we want to define a function field . over . and investigate its properties. We shall confine ourselves to the special case where . is algebraic, which is substantially easier and which will suffice for most of the applications we have in mind. H

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Drinfeld Modules: An Introduction,r properties. By considering the Carlitz module, in particular, we were able to construct a family of field extensions of . with properties remarkably similar to those of cyclotomic fields. In this chapter we will give a more general definition of a Drinfeld module. The definition and theory of thes

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The Brumer-Stark Conjecture,t the structure of the class group of cyclotomic number fields. This important conjecture, due to A. Brumer and H. Stark, is unresolved in the number field case. The analogous conjecture in function fields is now a theorem due to the efforts of J. Tate and P. Deligne. A short time after Deligne comp

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The Class Number Formulas in Quadratic and Cyclotomic Function Fields,umber fields. Some of these go back to the nineteenth century, e.g., the work of Dirichlet and Kummer. More recent contributions are associated with the names of Carlitz, Iwasawa, and Sinnott. We will review some of these results and then formulate and prove a number of analogues in the function fie

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Asymptomatic