  Hexagon relations are combinatorial or algebraic realizations of
four-dimensional Pachner moves. We introduce some simple set-theoretic hexagon
relations and then `quantize' them using what we call `polynomial hexagon
cohomologies'. Based on this, topological quantum field theories are proposed
with polynomial `discrete Lagrangian densities' taking values in finite fields.
First calculations of the resulting manifold invariants, arising from
polynomial cocycles of degree three and in characteristic two, show their
nontriviality.
