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