If $1+\mathrm{w}+{\mathrm{w}}^2=0$ and ${\mathrm{w}}^3=1$ then 

$(1-\mathrm{w}+{\mathrm{w}}^2)(1-{\mathrm{w}}^2+{\mathrm{w}}^4)(1-{\mathrm{w}}^4+{\mathrm{w}}^8)(1-{\mathrm{w}}^8+{\mathrm{w}}^{16})=$

$\begin{array}{rcl}{\mathrm{w}}^4& =& {\mathrm{w}}^3·\mathrm{w}=\mathrm{w}, {\mathrm{w}}^8={\left({\mathrm{w}}^3\right)}^2 {\mathrm{w}}^2={\mathrm{w}}^2, {\mathrm{w}}^{16}={\left({\mathrm{w}}^3\right)}^5·\mathrm{w}=\mathrm{w}\end{array}$

$\begin{array}{rcl} & & \left(1-\mathrm{w}+{\mathrm{w}}^2\right) \left(1-{\mathrm{w}}^2+{\mathrm{w}}^4\right) \left(1-{\mathrm{w}}^4+{\mathrm{w}}^8\right) \left(1-{\mathrm{w}}^8+{\mathrm{w}}^{16}\right)\\ & & \\ & & \left(1-\mathrm{w}+{\mathrm{w}}^2\right) \left(1-{\mathrm{w}}^2+\mathrm{w}\right) \left(1-\mathrm{w}+{\mathrm{w}}^2\right) \left(1-{\mathrm{w}}^2+\mathrm{w}\right)\\ & =& {\left(\left(1+{\mathrm{w}}^2+\mathrm{w}\right) \left(1+\mathrm{w}-{\mathrm{w}}^2\right)\right)}^2\\ & =& {\left(\left(-\mathrm{w}-\mathrm{w}\right) \left(-{\mathrm{w}}^2-{\mathrm{w}}^2\right)\right)}^2\\ & =& {\left(\left(-2\mathrm{w}\right) \left(-2{\mathrm{w}}^2\right)\right)}^2=4^2=16\end{array}$