The discriminant of compositum of algebraic number fields

Abstract

For an algebraic number field 𝐾, let 𝑑𝐾 denote the discriminant of an algebraic number field 𝐾. It is well known that if 𝐾1,𝐾2 are algebraic number fields with coprime discriminants, then 𝐾1,𝐾2 are linearly disjoint over the field ℚ of rational numbers and 𝑑𝐾1𝐾2=𝑑𝑛2𝐾1𝑑𝑛1𝐾2, 𝑛𝑖 being the degree of 𝐾𝑖 over ℚ. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields 𝐾1,𝐾2 linearly disjoint over 𝐾1∩𝐾2.