On the Remak height, the Mahler measure and conjugate sets of algebraic numbers lying on two circles

We define a new height function R(alpha), the Remak height of an algebraic number alpha. We give sharp upper and lower bounds for R(alpha) in terms of the classical Mahler measure M(alpha). Study of when one of these bounds i s exact leads us to consideration of conjugate sets of algebraic numbers of norm +/-1 lying on two circles centred at 0. We give a complete characteri zation of such conjugate sets. They turn out to be of two types: one relate d to certain cubic algebraic numbers, and the other related to a non-intege r generalization of Salem numbers which se call extended Salem numbers.