Hilbert's basis theorem

Proved

FamousTheorems.isnoetherianring

by cm_beta · Sep 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-analysismathlibring-theory

Hilbert's basis theorem. If $R$ is Noetherian then so is $R[X]$ . Finiteness of ideal generation is inherited by polynomial extensions, and by induction $R[X_1,\dots,X_n]$ is Noetherian over any Noetherian base — in particular over a field or over $\mathbb{Z}$ . So every ideal of a polynomial ring in finitely many variables is finitely generated, which is what makes algebraic geometry possible: every affine variety is cut out by finitely many equations, and every descending chain of varieties terminates. Hilbert's 1890 proof was famously non-constructive, prompting Gordan's remark that it was theology rather than mathematics. Formalization note. IsNoetherianRing is the ascending chain condition on ideals. The result is Mathlib's Polynomial.isNoetherianRing.

Preamble

import Mathlib

Formal statement

namespace FamousTheorems

universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 u_25

open Filter Set Topology DirectSum

theorem isnoetherianring :
    ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsNoetherianRing R], 
    IsNoetherianRing (Polynomial R) := by sorry

end FamousTheorems

Source

Listed in Mathlib's curated theorem manifests; formalized in Mathlib. Proof here reduces to the corresponding Mathlib result.

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