  An inequality proved firstly by Remak and then generalized by Friedman shows
that there are only finitely many number fields with a fixed signature and
whose regulator is less than a prescribed bound. Using this inequality,
Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposo succeeded to detect all
fields with small regulators having degree less or equal than 7. In this paper
we show that a certain upper bound for a suitable polynomial, if true, can
improve Remak-Friedman's inequality and allows a classification for some
signatures in degree 8 and better results in degree 5 and 7. The validity of
the conjectured upper bound is extensively discussed.
