A remark on Hilbert's Nullstellensatz.
Harley Flanders · Journal of the Mathematical Society of Japan · 1954
In [3] Zariski gave a proof of the Nullstellensatz based on the following lemma.Let $k$ and $K$ be fields such that $K=k[x_{1},\cdots, x_{n}]$ .Then $K$ is a finite extension of $k$ .Besides the proof of this lemma that Zariski gave, there is another proof in Artin and Tate [1] and a further proof is indicated in the exercises in Bourbaki [2, p. 87, exercise 4 and p. 106, exercise 12].