Counting generators of normal integral bases

Abstract. We present a very accurate formula for counting norms of normal integral bases in tame abelian extensions of the rational field. The methods used include applications of Schmidt’s Subspace Theorem, Baker’s Theorem and the Hardy-Littlewood Method, all from diophantine approximation. 1. Introduction. Let K denote a finite, tame, abelian Galois extension of the rational field Q with Γ denoting the Galois group. Let OK denote the ring of algebraic integers in K. Then there is an element a 2 OK whose conjugates under Γ form a Z-basis for OK. Such an element is known as a generator for a normal integral basis. It will be helpful to be able to express the existence of a norma