abstract algebra – Factor functions into multiplicative factors


Suppose we have a function $f(x_1 ,x_2 ,x_3 ,x_4).$ We know that we can factor it int two ways as $f(x_1 ,x_2 ,x_3 ,x_4)=phi_1 (x_1 ,x_2 )phi_2(x_3 ,x_4 )=psi_1 (x_1,x_3)psi_2(x_2,x_4)$

Show that we can completely factor the function as: $f(x_1 ,x_2 ,x_3 ,x_4)=varphi_1(x_1)varphi_2(x_2)varphi_3(x_3)varphi_4(x_4)$.

I’m sure this is true. This is just something I think of but cannot prove rigorously.