The Noether Normalization Lemma states that for any finitely generated commutative algebra over a field, there exist algebraically independent elements of that algebra such that the whole algebra becomes a finitely generated module over the polynomial ring they generate. Named for Emmy Noether, it is a foundational structural result of commutative algebra, giving every affine variety a finite map onto an affine space and underlying much of the geometric approach to ring theory.
Facts
StatementLet k be a field and A = k[y1', ..., ym'] be a finitely generated k-algebra. Then for some integer d, 0 <= d <= m, there exist y1, ..., yd in A algebraically independent over k such that A is finite over k[y1, ..., yd]. 1 Proof Method Classification
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Source Noether normalization lemma, Wikipedia
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Source Wikipedia: Emmy Noether
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1. Noether normalization lemma, Wikipedia
Lead section
In mathematics, the Noether normalization lemma (sometimes referred to as a theorem rather than a lemma) is a result of commutative algebra, introduced by Emmy Noether in 1926.
In Branch: Commutative Algebra, Lead sentence
as a theorem rather than a lemma) is a result of commutative algebra, introduced by Emmy Noether in 1926.
View the Source2. ProofWiki: Noether Normalization Lemma
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Since $A$ is finitely generated, we prove this by induction on the number $m$ of generators as a $k$-algebra.
Wikipedia: Emmy Noether
WikipediaPrincipal Result Of: Emmy Noether, Article textQuote, Principal Result Of: Emmy Noether, Article text
In this paper Noether also introduced the Noether normalization lemma, showing that a finitely generated domain A over a field k has a set {x1, ..., xn} of algebraically independent elements such that A is integral over k[x1, ..., xn].
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