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
StatementEvery ideal of a polynomial ring over a field has a finite generating set, called a finite basis in Hilbert's own terminology. 1 Proof Method Classification
Statement Form Connections
Has Statement Form
In Branch
Named After
Principal Result Of
Source David Hilbert (Wikipedia)
Proved By
Sources
1. Hilbert's Basis Theorem (Wikipedia)
Wikimedia FoundationLead 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
View the Source 2. ProofWiki: Hilbert's Basis Theorem
ProofWikiProof sectionQuote, Proof section
Aiming for a contradiction, suppose not.
David Hilbert (Wikipedia)
Principal Result Of: David Hilbert, Article textQuote, Principal Result Of: David Hilbert, Article text
As a result, he demonstrated Hilbert's basis theorem, showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.