Kummer′s Criterion over Global Function Fields Original Research Article

Let k be a global function field over the field imageq where q = pm and let ∞ be a fixed prime of k which we assume to have degree 1. We let A be the ring of functions holomorphic outside of ∞ and we let Weierstrass p be a prime of A. Hayes has associated to {A, Weierstrass p} a finite abelian extension k(Weierstrass p) of k which is analogous to the classical cyclotomic extension image(ζp), where ζp is a primitive pth root of unity. In this paper we define a certain set {Bi} of A-fractional ideals of k (i.e., a set of finite divisors), and we establish an analog of Kummer′s Theorem for {k(Weierstrass p), Bi}.