  In this paper we introduce a new algebraic method in tilings. Combining this
method with Hilbert's Nullstellensatz we obtain a necessary condition for
tiling $n$-space by translates of a cluster of cubes. Further, the polynomial
method will enable us to show that if there exists a tiling of $n$-space by
translates of a cluster $V$ of prime size then there is a lattice tiling by $V$
as well. Finally, we provide supporting evidence for a conjecture that each
tiling by translates of a prime size cluster $V$ is lattice if $V$ generates
$n$-space.
