  The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$
is a field and $G$ is a torsion-free group then the only units of the group
ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of
group elements. We give a concrete counterexample to this conjecture; the group
is virtually abelian and the field is order two.
