RATIOS OF REGULATORS IN TOTALLY-REAL EXTENSIONS OF NUMBER-FIELDS

Let K under(⊂, ≠) L be an extension of totally real number fields. We prove that their regulators satisfy Reg(L)>Reg(K), except for finitely many extensions L/K. We do this by showing Reg(L) > (1.28r / ((r + 2) sqrt(r))) Reg(K), where r=[L:Q]-[K:Q], and Reg(L)/Reg(K)>C log {norm of matrix}DL{norm of matrix}, where DL is the absolute discriminant of L and >0 depends only on [L:Q]. © 1991 Academic Press, Inc.