Lift the chosen binary polynomial f to the same monic polynomial over the integers. Its irreducibility modulo two implies irreducibility over Q, so K=Q[U]/(f) is a field. Set O=Z_(2)[U]/(f). Every element has e coordinates in the basis 1,U,...,U^(e-1); define its valuation as the minimum coordinate valuation. For nonzero x,y, factor out those minima. The remaining elements have nonzero residues in the field F_(2^e), so their product also has nonzero residue. Consequently v_2(xy)=v_2(x)+v_2(y). This argument needs the irreducible residue polynomial.