= Solution
<Zariski lemma> says that a field which is a <finitely generated algebra> over a field $k$ is a finite algebraic extension of $k$. The <Strong Hilbert Nullstellensatz> says that, for an ideal $I\subseteq k[T_1,\ldots,T_n]$ over an <algebraically closed field>,
$$
I(V(I))=\sqrt I.
$$
Let the unique point of $V(\mathfrak a)$ be $x=(x_1,\ldots,x_n)$ and let
$$
\mathfrak m=(T_1-x_1,\ldots,T_n-x_n).
$$
The Nullstellensatz gives $\sqrt{\mathfrak a}=\mathfrak m$, so $\mathfrak a\subseteq\mathfrak m$. Each of the finitely many generators $u_i=T_i-x_i$ of $\mathfrak m$ has some power $u_i^{e_i}\in\mathfrak a$. If
$$
r=1+\sum_i(e_i-1),
$$
then every monomial of total degree $r$ in the $u_i$ is divisible by one of the $u_i^{e_i}$. Hence
$$
\mathfrak m^r\subseteq\mathfrak a\subseteq\mathfrak m.
$$
Thus the assertion is true; algebraically, the quotient defines a <punctual scheme> supported at $x$.
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