---
tag: 7Z28JY8QW4
kind: statement
role: theorem
name: Noether normalization lemma
msc: [13B21, "13E15"]
uses: [04C5ZBG1WT, AC5TFFQT9M, 1NBACAEC2V, 0SDC934R52, D7TR0YWR9H, MVGBS3J1P7]
about: [AC5TFFQT9M, B7X0KFY1S1]
sources:
  - url: https://en.wikipedia.org/wiki/Noether_normalization_lemma
same_as:
  wikidata: Q1287074
  wikipedia: Noether normalization lemma
---

## Statement

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a nonzero [[finitely generated|AC5TFFQT9M]] $k$-algebra.
Then there are an integer $d \ge 0$ and elements $y_1, \ldots, y_d \in A$ such that:

1. $y_1, \ldots, y_d$ are algebraically independent over $k$, so the subalgebra $k[y_1, \ldots, y_d]$ is a [[polynomial ring|0SDC934R52]] in $d$ variables.
2. $A$ is a [[finite|1NBACAEC2V]] $k[y_1, \ldots, y_d]$-algebra, that is, a finitely generated module over it.

## Remarks

The proof is by induction on the number of generators of $A$.
Suppose the generators $x_1, \ldots, x_n$ satisfy a nonzero polynomial relation.
The substitution $x_i' = x_i - x_n^{r^i}$ for $i < n$, with $r$ large, turns the relation into one that is monic in $x_n$ up to a nonzero constant.
Then $A$ is finite over $k[x_1', \ldots, x_{n-1}']$, and one continues with this smaller algebra.
When $k$ is infinite, linear substitutions $x_i' = x_i - c_i x_n$ with suitable $c_i \in k$ suffice.

The number $d$ is the [[Krull dimension|MVGBS3J1P7]] of $A$, by [[going-up|GHKYR5PBQC]] and [[incomparability|Q2XBHJ96GV]].
When $A$ is a domain, $d$ is also the [[transcendence degree|D7TR0YWR9H]] of its field of fractions over $k$.

Example: let $A = k[x, y]/(xy - 1)$ and $t = x + y$.
In $A$, $x(t - x) = xy = 1$, so $x^2 - t x + 1 = 0$ and likewise for $y$; hence $A$ is generated by $1$ and $x$ as a $k[t]$-module.
The element $t$ is transcendental over $k$, since $A$ is not finite-dimensional over $k$, so $d = 1$.
Note that $x$ alone does not work: $A = k[x, x^{-1}]$ is not finite over $k[x]$.

## Explanations

### L1 human

Every finitely generated algebra over a field sits finitely over a polynomial ring.
Geometrically, every affine variety of dimension $d$ maps onto the $d$-dimensional affine space by a map with finite fibres.
So complicated varieties can be studied as finite covers of the simplest ones.

### L2 human

Noether normalization is the main tool to compute the dimension of finitely generated algebras over a field and to prove that it equals the transcendence degree.
It gives a short proof of [[Zariski's lemma|YBJ2VQHFDQ]] and hence of the [[weak Nullstellensatz|ZWS0YWGF17]].
If $A$ is a field, then $k[y_1, \ldots, y_d]$ is a field by [[integral extensions of fields|5GCANV7337]], so $d = 0$.
It is also used to show that finitely generated domains over a field are catenary.