Application
The normalization lemma can be used to show: for a finitely generated k-algebra A,
(the transcendental degree of an integral domain is that of the field of fractions of the domain.) Indeed, by the normalization lemma, A is integral over the polynomial ring . The formula above is clear for S. But then A and S have the same Krull dimension by integrality, and A and S have the same transcendental degree since the field of fractions of A is algebraic over that of S. The general formula now follows.
The formula says in particular that a field that is a finitely generated k-algebra is a finite field extension of k. This is known as Zariski's lemma.
Read more about this topic: Noether Normalization Lemma
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