Distribution of units of a cubic abelian field modulo prime numbers

Abstract

We studied the distribution of units of an algebraic number field modulo prime ideals. Here we study the distribution of units of a cubic abelian field modulo rational prime numbers. For a decomposable prime number $p$, $2(p-1{)}^2$ is an upper bound of the order of the unit group modulo $p$, and we show that the conjectural density of primes which attain it is really positive.