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  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be a nonzero [[finitely generated|AC5TFFQT9M]] $k$-algebra.\nThen there are an integer $d \\ge 0$ and elements $y_1, \\ldots, y_d \\in A$ such that:\n\n1. $y_1, \\ldots, y_d$ are algebraically independent over $k$, so the subalgebra $k[y_1, \\ldots, y_d]$ is a [[polynomial ring|0SDC934R52]] in $d$ variables.\n2. $A$ is a [[finite|1NBACAEC2V]] $k[y_1, \\ldots, y_d]$-algebra, that is, a finitely generated module over it.\n\n## Remarks\n\nThe proof is by induction on the number of generators of $A$.\nSuppose the generators $x_1, \\ldots, x_n$ satisfy a nonzero polynomial relation.\nThe substitution $x_i' = x_i - x_n^{r^i}$ for $i \u003c n$, with $r$ large, turns the relation into one that is monic in $x_n$ up to a nonzero constant.\nThen $A$ is finite over $k[x_1', \\ldots, x_{n-1}']$, and one continues with this smaller algebra.\nWhen $k$ is infinite, linear substitutions $x_i' = x_i - c_i x_n$ with suitable $c_i \\in k$ suffice.\n\nThe number $d$ is the [[Krull dimension|MVGBS3J1P7]] of $A$, by [[going-up|GHKYR5PBQC]] and [[incomparability|Q2XBHJ96GV]].\nWhen $A$ is a domain, $d$ is also the [[transcendence degree|D7TR0YWR9H]] of its field of fractions over $k$.\n\nExample: let $A = k[x, y]/(xy - 1)$ and $t = x + y$.\nIn $A$, $x(t - x) = xy = 1$, so $x^2 - t x + 1 = 0$ and likewise for $y$; hence $A$ is generated by $1$ and $x$ as a $k[t]$-module.\nThe element $t$ is transcendental over $k$, since $A$ is not finite-dimensional over $k$, so $d = 1$.\nNote that $x$ alone does not work: $A = k[x, x^{-1}]$ is not finite over $k[x]$.\n\n## Explanations\n\n### L1 human\n\nEvery finitely generated algebra over a field sits finitely over a polynomial ring.\nGeometrically, every affine variety of dimension $d$ maps onto the $d$-dimensional affine space by a map with finite fibres.\nSo complicated varieties can be studied as finite covers of the simplest ones.\n\n### L2 human\n\nNoether 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.\nIt gives a short proof of [[Zariski's lemma|YBJ2VQHFDQ]] and hence of the [[weak Nullstellensatz|ZWS0YWGF17]].\nIf $A$ is a field, then $k[y_1, \\ldots, y_d]$ is a field by [[integral extensions of fields|5GCANV7337]], so $d = 0$.\nIt is also used to show that finitely generated domains over a field are catenary.\n",
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