Noether normalization theorem

Statement

Let $k$ be a field and $R$ a finitely generated $k$-algebra. Let $P_r\ge P_{r-1}\ge \dots \ge P_0$ be a chain of descending ]]prime ideal]]s, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring $S$ of $R$ with $S\cong k[x_1,x_2,\dots ,x_r]$ sich that $R$ is a finitely generated $S$-module and $P_i\cap S=(x_1,x_2,\dots ,x_i)$.

Applications

• Hilbert's nullstellensatz