# Zariski's lemma

In algebra, Zariski's lemma, introduced by Oscar Zariski, 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 ${\displaystyle k[t_{1},...,t_{n}]}$ (k algebraically closed field), then I has a zero; i.e., there is a point x in ${\displaystyle k^{n}}$ such that ${\displaystyle f(x)=0}$ 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 ${\displaystyle k[x_{1},\ldots ,x_{d}]}$ where ${\displaystyle x_{1},\ldots ,x_{d}}$ are algebraically independent over k. But since K has Krull dimension zero, the polynomial ring must have dimension zero; i.e., ${\displaystyle d=0}$. For Zariski's original proof, see the original paper.[6]

In fact, the lemma is a special case of the general formula ${\displaystyle \dim A=\operatorname {tr.deg} _{k}A}$ for a finitely generated k-algebra A that is an integral domain, which is also a consequence of the normalization lemma.

## Notes

1. ^ Milne, Theorem 2.6
2. ^ Proof: it is enough to consider a maximal ideal ${\displaystyle {\mathfrak {m}}}$. Let ${\displaystyle A=k[t_{1},...,t_{n}]}$ and ${\displaystyle \phi :A\to A/{\mathfrak {m}}}$ be the natural surjection. By the lemma, ${\displaystyle A/{\mathfrak {m}}=k}$ and then for any ${\displaystyle f\in {\mathfrak {m}}}$,
${\displaystyle f(\phi (t_{1}),\cdots ,\phi (t_{n}))=\phi (f(t_{1},\cdots ,t_{n}))=0}$;
that is to say, ${\displaystyle x=(\phi (t_{1}),\cdots ,\phi (t_{n}))}$ is a zero of ${\displaystyle {\mathfrak {m}}}$.
3. ^ Atiyah-MacDonald 1969, Ch 5. Exercise 25
4. ^ Atiyah–MacDonald 1969, Ch 5. Exercise 18
5. ^ Atiyah–MacDonald 1969, Proposition 7.9