Theorem

Hilbert's Basis Theorem

Algebra

If a ring is Noetherian, meaning every ideal in it is finitely generated, then the polynomial ring formed over it in any finite number of variables is also Noetherian. Proved by David Hilbert, it is foundational to commutative algebra and algebraic geometry.

Facts

Statement

Every ideal of a polynomial ring over a field has a finite generating set, called a finite basis in Hilbert's own terminology. 1

Proof Year

1890 1

Classification

Statement Form

Existence Theorem 1

Connections

Sources

1. Hilbert's Basis Theorem (Wikipedia)

Wikimedia Foundation

Lead section, opening sentence
Hilbert's basis theorem asserts that every ideal of a polynomial ring over a field has a finite generating set (a finite basis in Hilbert's terminology).
History note on the 1890 invariant theory paper
The theorem was stated and proved by David Hilbert in 1890 in his seminal article on invariant theory, where he solved several problems on invariants.
In Group: Ring Theory and Commutative Algebra, Lead section
of the interpretation of algebraic geometry in terms of commutative algebra. In particular, the basis theorem implies that

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Hilbert's Basis Theorem

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