How to show that the dimension of a quotient space in the field of polynomials is not finite?

Question

I have to show that if I have a quotient of the form $\mathbb{K}[x_1,x_2,\dots,x_n]/\langle f_1,f_2,\dots,f_s\rangle$, $\operatorname{char}(\mathbb{K})\not=0$, and on which the class of $[x_i^l]$ is not zero for every $i\in\{1,2,\dots,n\}$ and for all $l>0$, then said space can't have finite dimension. The problem is I can't find a proof that in that case said monomials will be independent, supposing they really are. Any help?