# 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.