Units in residue classes (Q1100508)

It is shown that if \(K\neq {\mathbb{Q}}\) is a real Abelian field of degree \(>3\), then there exist infinitely many prime ideals P of first degree in K/\({\mathbb{Q}}\) such that every non-zero residue class (mod P) contains a unit. The same holds also for real quadratic and cubic Abelian fields with at most two exceptions.