# Zariski lemma

###### Proposition 1.

Let $R\mathrm{\subseteq}S\mathrm{\subseteq}T$ be commutative rings. If $R$ is noetherian^{}, and T finitely generated^{} as an $R$-algebra^{} and as an $S$-module, then $S$ is finitely generated as an $R$-algebra.

###### Lemma 1 (Zariski’s lemma).

Let $\mathrm{(}L\mathrm{:}K\mathrm{)}$ be a field extension and ${a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{a}_{n}\mathrm{\in}L$ be such that $K\mathit{}\mathrm{(}{a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{a}_{n}\mathrm{)}\mathrm{=}K\mathit{}\mathrm{[}{a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{a}_{n}\mathrm{]}$. Then the elements ${a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{a}_{n}$ are algebraic over $K$.

###### Proof.

The case $n=1$ is clear. Now suppose $n>1$ and not all ${a}_{i},1\le i\le n$ are algebraic over $K$.

Wlog we may assume ${a}_{1},\mathrm{\dots},{a}_{n}$ are algebraically independent^{} and each element ${a}_{r+1},\mathrm{\dots},{a}_{n}$ is algebraic over $D:=K({a}_{1},\mathrm{\dots},{a}_{r})$. Hence $K[{a}_{1},\mathrm{\dots},{a}_{n}]$ is a finite algebraic extension^{} of $D$ and therefore is a finitely generated $D$-module.

The above proposition^{} applied to $K\subseteq D\subseteq K[{a}_{1},\mathrm{\dots},{a}_{n}]$ shows that $D$ is finitely generated as a $K$-algebra, i.e $D=K[{d}_{1},\mathrm{\dots},{d}_{n}]$.

Let ${d}_{i}=\frac{{p}_{i}({a}_{1},\mathrm{\dots},{a}_{n})}{{q}_{i}({a}_{1},\mathrm{\dots},{a}_{n})}$, where ${p}_{i},{q}_{i}\in K[{x}_{1},\mathrm{\dots},{x}_{n}]$.

Now ${a}_{1},\mathrm{\dots},{a}_{n}$ are algebraically independent so that $K[{a}_{1},\mathrm{\dots},{a}_{n}]\cong K[{x}_{1},\mathrm{\dots},{x}_{n}]$, which is a UFD (http://planetmath.org/UFD).

Let $h$ be a prime divisor^{} of ${q}_{1}\mathrm{\cdots}{q}_{r}+1$. Since $q$ is relatively prime to each of ${q}_{i}$, the element $q{({a}_{1},\mathrm{\dots},{a}_{n})}^{-1}\in D$ cannot be in $K[{d}_{1},\mathrm{\dots},{d}_{n}]$. We obtain a contradiction^{}.
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Title | Zariski lemma |
---|---|

Canonical name | ZariskiLemma |

Date of creation | 2013-03-22 17:18:11 |

Last modified on | 2013-03-22 17:18:11 |

Owner | polarbear (3475) |

Last modified by | polarbear (3475) |

Numerical id | 7 |

Author | polarbear (3475) |

Entry type | Derivation^{} |

Classification | msc 12F05 |

Classification | msc 11J85 |