diego@walsh:~$ Singular 
                     SINGULAR                             /  Development
 A Computer Algebra System for Polynomial Computations   /   version 3-0-4
                                                       0<
     by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   Nov 2007
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
> ring A=(2^4,b),x,lp;
> poly m=(x-b)*(x-b^2)*(x-b^4)*(x-b^8);
> m;
x4+x+1
> poly mm=(x-b^3)*(x-b^6)*(x-b^12)*(x-b^9);
> mm;
x4+x3+x2+x+1
> poly mmm=(x-b^7)*(x-b^14)*(x-b^13)*(x-b^11);
> mmm;
x4+x3+1
> poly mmmm=(x-b^5)*(x-b^10);
> mmmm;
x2+x+1
> poly mmmmm=(x-b^0);
> mmmmm;
x+1
> m*mm*mmm*mmmm*mmmmm;
x15+1