= Solution
Let $\mathfrak m$ be maximal in $\mathbb Q[T_1,\ldots,T_n]$ and put
$$
K=\mathbb Q[T_1,\ldots,T_n]/\mathfrak m.
$$
The <Zariski lemma> makes $K$ a finite extension of $\mathbb Q$. Base change gives
$$
\mathbb C[T_1,\ldots,T_n]/\mathfrak m\mathbb C[T_1,\ldots,T_n]
\cong\mathbb C\otimes_{\mathbb Q}K.
$$
This ring is nonzero because the field extension makes $\mathbb C$ a <faithfully flat module> over $\mathbb Q$. Choose a maximal ideal $\mathfrak n$ of the quotient, or equivalently a maximal ideal of $\mathbb C[T_1,\ldots,T_n]$ containing $\mathfrak m\mathbb C[T_1,\ldots,T_n]$. Its contraction to the rational polynomial ring contains $\mathfrak m$ and is proper, so maximality of $\mathfrak m$ forces
$$
\mathfrak n\cap\mathbb Q[T_1,\ldots,T_n]=\mathfrak m.
$$
Solved by gpt-5.6-sol high.
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