Noether normalization lemma

For a finitely generated algebra over a field, there is a polynomial subalgebra generated by algebraically independent elements over which the algebra is a finite module. It gives an algebraic version of a finite map to affine space.

Integral extension: A ring extension in which every element of the larger ring satisfies a monic polynomial over the smaller ring. The normalization proof makes the algebra integral over its polynomial subalgebra.
Finitely generated algebra: An algebra generated by finitely many elements over a base ring. Finite generation is the lemma’s starting condition.
Affine variety: A geometric object defined by polynomial equations over a field. For coordinate rings, normalization yields a finite projection from the variety to affine space.
Emmy Noether: A German mathematician whose work reshaped abstract algebra and mathematical physics. The lemma bears her name and reflects her influence on modern ring theory.
Finite morphism: A morphism of schemes that is affine and whose coordinate rings are finite modules over one another. Module finiteness translates the lemma into a finite map of affine varieties.
Polynomial ring: A ring of polynomials in one or more indeterminates with coefficients in a specified ring. The subalgebra guaranteed by the lemma is a polynomial ring.
Noetherian normalization: A theorem asserting that a finitely generated algebra over a field is integral over a polynomial subalgebra. This common name emphasizes the integral-extension form used in geometric applications.
Commutative algebra: The study of commutative rings, their ideals, and modules over them. Noether normalization became a foundational structural theorem in this field.
Transcendence basis: A maximal algebraically independent subset over which a field extension is algebraic. The polynomial parameters supplied by normalization have the size of a transcendence basis.
Algebraic independence: A set of elements is algebraically independent over a field if no nonzero polynomial over that field vanishes on them. The normalization parameters must satisfy precisely this condition.

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Lüroth's theoremCompared with: It gives polynomial coordinates in broader settings but does not guarantee an intermediate field is purely transcendental.