Small points and free abelian groups

Abstract: Let $F$ be an algebraic extension of the rational numbers and $E$ an elliptic curve defined over some number field contained in $F$. The absolute logarithmic Weil height, respectively the N\'eron-Tate height, induces a norm on $F^*$ modulo torsion, respectively on $E(F)$ modulo torsion. The groups $F^*$ and $E(F)$ are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. In this paper we prove the failure of the converse to this statement by explicitly constructing counterexamples.