On normal integral bases of unramified abelian p-extensions over a global function field of characteristic p

Abstract

Let K be an algebraic function field of one variable over a finite field of characteristic p, and S a finite non-empty set of prime divisors of K. As the ring of integers of K, we take the ring of elements of K integral outside S. We prove that for a finite abelian p-extension L/ K, it has a relative normal integral basis (NIB) if and only if it is unramified outside S. We also give a generator of NIB in an explicit form.