# Zariski's lemma

In algebra, **Zariski's lemma**, introduced by Oscar Zariski (1947), states that if *K* is a finitely generated algebra over a field *k* and if *K* is a field, then *K* is a finite field extension of *k*.

An important application of the lemma is a proof of the weak form of Hilbert's nullstellensatz:^{[1]} if *I* is a proper ideal of (*k* algebraically closed field), then *I* has a zero; i.e., there is a point *x* in such that for all *f* in *I*.^{[2]}

The lemma may also be understood from the following perspective. In general, a ring *R* is a Jacobson ring if and only if every finitely generated *R*-algebra that is a field is finite over *R*.^{[3]} Thus, the lemma follows from the fact that a field is a Jacobson ring.

## Proof

Two direct proofs, one of which is due to Zariski, are given in Atiyah–MacDonald.^{[4]}^{[5]} The lemma is also a consequence of the Noether normalization lemma. Indeed, by the normalization lemma, *K* is a finite module over the polynomial ring where are algebraically independent over *k*. But since *K* has Krull dimension zero, the polynomial ring must have dimension zero; i.e., . For Zariski's original proof, see the original paper.^{[6]}

In fact, the lemma is a special case of the general formula for a finitely generated *k*-algebra *A* that is an integral domain, which is also a consequence of the normalization lemma.

## Notes

- ↑ Milne, Theorem 2.6
- ↑ Proof: it is enough to consider a maximal ideal . Let and be the natural surjection. By the lemma, and then for any ,
- ;

- ↑ Atiyah-MacDonald 1969, Ch 5. Exercise 25
- ↑ Atiyah–MacDonald 1969, Ch 5. Exercise 18
- ↑ Atiyah–MacDonald 1969, Proposition 7.9
- ↑ http://projecteuclid.org/euclid.bams/1183510605

## References

- M. Atiyah, I.G. Macdonald,
*Introduction to Commutative Algebra*, Addison–Wesley, 1994. ISBN 0-201-40751-5 - James Milne, Algebraic Geometry
- Zariski, Oscar (1947), "A new proof of Hilbert's Nullstellensatz",
*Bull. Amer. Math. Soc.*,**53**: 362–368, MR 0020075