Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 3 / a

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 $I\subseteq k[T_1,\dots ,T_n]$ over an algebraically closed field,

Let the unique point of $V(\mathfrak{a})$ be $x=(x_1,\dots ,x_n)$ and letThe 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 $u_i^{e_i}\in \mathfrak{a}$. Ifthen every monomial of total degree $r$ in the $u_i$ is divisible by one of the $u_i^{e_i}$. HenceThus the assertion is true; algebraically, the quotient defines a punctual scheme supported at $x$.