Maximal independent systems of units in global function fields
Fei Xu, Jianqiang Zhao · Acta Arithmetica · 1996
Introduction.Around 1980, Galovich and Rosen (cf.[GR1] and [GR2]) computed the index of cyclotomic units in the full group of units in a cyclotomic function field over a rational function field over a finite field.Later, Hayes [H1] and Oukhaba [O] obtained a few index formulae of the elliptic units in some special extensions of the global function fields with some restrictions on the infinite prime.Feng and Yin [FY] constructed the maximal independent systems of cyclotomic units in finite abelian extensions over a rational function field.Recently Shu [S] extended the result in [GR1] to the global function fields with degree one infinite prime.In the Mini-Conference on the Arithmetic of Function Fields held at Brown University in April 1996, Yin announced a result along this line which extended his work in [Y] by removing the restriction on the degree of the infinite prime.This result seems to be the best one concerning the cyclotomic unit index in the sense that the base field can be any global function field.In this paper we construct maximal independent systems of units in an abelian extension over such a global function field.Notations and terminology are standard if not explained.Specifically,• k: a global function field with a constant field F q of q elements.• ∞: a fixed infinite prime.• A: the ring of the functions in k which are holomorphic away from ∞.• M ∞ : the set of integral ideals of A.• e: the unit ideal of A.• P: the set of k-primes (k-places).• k v : the local field over k completed at any v ∈ P.For any m ∈ M ∞ ,• P m := {p ∈ P : (p, m) = 1}.• M m := {b ∈ M ∞ : (b, m) = 1}.• m v (resp.A v ): the completion of m (resp.A) in k v .