S-Units, S-Class Group, and the Corresponding L-Functions
Let K/F be an algebraic function field over the field of constants F. Throughout this book we have been emphasizing the analogy between the arithmetic of K and that of an algebraic number field. This analogy is particulary clear when we choose an element x ∈ K which is not a constant. The ring A = F[x] ⊂ k = F(x) then plays the role of the pair ℤ ⊂ ℚ in number theory. K is an algebraic extension of F(x) and the analogue of the ring of integers in an algebraic number field is the integral closure of A in K. Let’s call this ring B. We will show that B is a Dedekind domain. We will investigate the unit group and the class group of B. We will also associate zeta and L-functions to B.
KeywordsGalois Group Function Field Class Number Integral Closure Effective Divisor
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