The smallest prime that splits completely in an abelian number field

Abstract

Let K / Q K/\mathbf {Q} be an abelian extension and let D D be the absolute value of the discriminant of K K . We show that for each ε > 0 \varepsilon > 0 , the smallest rational prime that splits completely in K K is O ( D 1 4 + ε ) O(D^{\frac 14+\varepsilon }) . Here the implied constant depends only on ε \varepsilon and the degree of K K . This generalizes a theorem of Elliott, who treated the case when K / Q K/\mathbf {Q} has prime conductor.