RATIOS OF REGULATORS IN EXTENSIONS OF NUMBER-FIELDS

Let L/K be an extension of number fields. Then Reg(L)/Reg(K) > c[L:Q](log\D(L)\)m, where Reg denotes the regulator, D(L) is the absolute discriminant of L , and c[L . Q] > 0 depends only on the degree of L . The nonnegative integer m = m(L/K) is positive if L/K does not belong to certain precisely defined infinite families of extensions, analogous to CM fields, along which Reg(L)/ Reg(K) is constant. This generalizes some inequalities due to Remak and Silverman, who assumed that K is the rational field Q, and modifies those of Berge-Martinet, who dealt with a general extension L/K but used its relative discriminant where we use the absolute one.