Hilbert's Basis Theorem

Hilbert's basis theorem states that the polynomial ring over a Noetherian ring is Noetherian, ensuring that systems of polynomial equations can always be described by finitely many generators.

Statement

Theorem4.3Hilbert's basis theorem

If $R$ is a Noetherian ring, then $R[x]$ is Noetherian. By induction, $R[x_1, \ldots, x_n]$ is Noetherian for all $n \geq 0$ .

Proof

Let $I \subseteq R[x]$ be an ideal. We show $I$ is finitely generated.

For each $n \geq 0$ , let $L_n = \{a \in R : \exists f \in I \text{ with } \deg(f) = n \text{ and leading coefficient } a\} \cup \{0\}$ . Then $L_n$ is an ideal of $R$ (if $a, b$ are leading coefficients of $f, g \in I$ of degree $n$ , then $a - b$ is the leading coefficient of $f - g$ or has lower degree; and $ra$ is the leading coefficient of $rf$ ).

Moreover, $L_0 \subseteq L_1 \subseteq L_2 \subseteq \cdots$ (if $f$ has degree $n$ and leading coefficient $a$ , then $xf$ has degree $n+1$ and leading coefficient $a$ , so $L_n \subseteq L_{n+1}$ ).

Since $R$ is Noetherian, the ascending chain $L_0 \subseteq L_1 \subseteq \cdots$ stabilizes: there exists $N$ with $L_n = L_N$ for all $n \geq N$ . Each $L_n$ (for $0 \leq n \leq N$ ) is finitely generated as an ideal of $R$ : say $L_n = (a_{n,1}, \ldots, a_{n,k_n})$ .

For each generator $a_{n,j}$ , choose $f_{n,j} \in I$ of degree $n$ with leading coefficient $a_{n,j}$ .

Claim: $I = (f_{n,j} : 0 \leq n \leq N,\, 1 \leq j \leq k_n)$ .

Let $g \in I$ with $\deg(g) = d$ . If $d \geq N$ : the leading coefficient of $g$ lies in $L_d = L_N$ , so it is an $R$ -linear combination of the $a_{N,j}$ . Subtracting the appropriate combination of $x^{d-N}f_{N,j}$ , we obtain an element of $I$ of degree $< d$ . If $d < N$ : similarly subtract combinations of $f_{d,j}$ to reduce the degree.

By induction on degree, $g$ is in the ideal generated by the $f_{n,j}$ . $\blacksquare$

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Consequences

ExampleConsequences of the basis theorem

$k[x_1, \ldots, x_n]$ is Noetherian for any field $k$ : every ideal of polynomials is finitely generated. This is the algebraic content of Hilbert's theorem on invariants.
Nullstellensatz prerequisite: Noetherianity of polynomial rings is essential for the Hilbert Nullstellensatz and the finite generation of radical ideals.
Groebner bases: The algorithm for computing Groebner bases terminates precisely because polynomial ideals are finitely generated (Noetherianity guarantees the ascending chain condition on leading term ideals).

RemarkHistorical significance

Hilbert proved this theorem in 1888 to show that the ring of invariants of a group action on polynomials is finitely generated. Gordan, who had proved finiteness for binary forms by explicit construction, reportedly said "This is not mathematics, it is theology" -- referring to the non-constructive nature of Hilbert's proof, which shows finite generation without producing explicit generators.