$\mathbb{Z}[1/3]$ is not finitely generated as a $\mathbb{Z}$-module.

Question

I wan't to show that $\mathbb{Z}[1/3]$ is not finitely generated as a $\mathbb{Z}$-module. So I suppose toward a contradiction that it is finitely generated, i.e. $\mathbb{Z}[1/3]=(x_1,...,x_d)$ where $x_1=p_1(1/3),...,x_d=p_d(1/3)$ i.e. they are polynomial expressions (with integral coefficients) in $1/3$, but then what element can't I express with linear combination of these elements, or where would there be a contradiction?