Show that $\Bbb Q$ is not a finitely generated $\Bbb Z$-algebra. Show that $\Bbb Q$ is not a finitely generated $\Bbb Z$-algebra.


I know that $\Bbb Q$ is not a finitely generated $\Bbb Z$-module. From here how can I conclude that $\Bbb Q$ is not a finitely generated $\Bbb Z$-algebra?

Please help me in this regard. 

Thank you in advance.
Hint:

If we had $\mathbf Q=\mathbf Z\biggl[\dfrac{m_1}{n_1},\dots,\dfrac{m_k}{n_k} \biggr]$, we could as well write
$$\mathbf Q=\mathbf Z\biggl[\frac1N \biggr],\ \text{where }\;N=\operatorname{lcm}(n_1,\dots,n_k).$$
Consider a prime $p$ not dividing $N$. Can $\dfrac1p$ be a polynomial in
$\dfrac 1N$?
Suppose $\mathbb Q$ is finitely generated as a $\mathbb Z$-module (in this case, $\mathbb Z$-algebra). By the Structure Theorem for modules (in this case, it is the theorem on finitely-generated abelian groups) and the fact that $\mathbb Q$ is torsion free, $\mathbb Q\cong \mathbb Z^r$ for some $r\geq 0$. Let $\varphi:\mathbb Q\rightarrow \mathbb Z^r$ be an isomorphism. Since $\varphi$ is a $\mathbb Z$-module homomorphism, if $n\in \mathbb Z$ is nonzero, then $\varphi(1)=\varphi\left(n\cdot \frac{1}{n}\right)=n\varphi\left(\frac{1}{n}\right)$. Let $\varphi(1)=(a_1,\ldots,a_r)$. Take $n=\mbox{gcd}(a_1,\ldots,a_r)+1$. Then $\varphi\left(\frac{1}{n}\right)\notin\mathbb Z^r$, a contradiction.