We investigate the values of the Remak height, which is a weighted product of the conjugates of an algebraic number. We prove
that the ratio of logarithms of the Remak height and of the Mahler measure for units αof degree d is everywhere dense in the maximal interval [d/2(d-1),1] allowed for this ratio. To do this, a “large” set of totally positive Pisot units is constructed. We also give a lower
bound on the Remak height for non-cyclotomic algebraic numbers in terms of their degrees. In passing, we prove some results
about some algebraic numbers which are a product of two conjugates of a reciprocal algebraic number.
. In this paper we discuss the construction of metric heights in general. More specifically, we study in some detail the metric
height obtained from the na"ve height of an algebraic number (the maximum modulus of the coefficients of its minimal polynomial).
In particular, we compute this metric height for some classes of surds.