  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.
