A polynomial ring in finitely many indeterminates over an integral domain is an integral domain

Depends on

• Polynomial rings in finitely many commuting indeterminates by iteration
• A polynomial ring over an integral domain is an integral domain
• The principle of mathematical induction

Used by

• A nonradical ideal need not enlarge every tangent space Counterexample
• For a field F, the ideal (x,y) in F[x,y] is not principal Counterexample
• Monomials, coefficients, degree in each variable and total degree in F[x₁,…,xₙ] Definition
• Multiplicity of a hypersurface equation at a rational point Definition
• A nondegenerate projective quadric Example
• A projective cone with a smooth conic base Example
• An irreducible curve can have arbitrarily large tangent dimension Example
• Different singularities can share a tangent cone Example
• Fitting ideals of a diagonal two-by-two presentation Example
• Localizing (x²,xy)=(x)∩(x,y)² keeps only the matching component Example
• Normalization of a nodal affine plane curve Example
• The cusp retains a doubled tangent line Example
• The diagonal bimodule k[x] is projective on both sides but not over its enveloping algebra Example
• The subring k[x,y,x/y,x/y²,…] of k(x,y) has a strictly ascending chain of principal ideals Example
• A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial Lemma
• All initial forms define the tangent cone Lemma
• Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields Lemma
• regular local graded surjection has zero kernel Lemma
• The Picard group and intersection form of a product of projective lines Lemma
• The Picard group of a point blowup of the projective plane Lemma
• The product of two projective lines is an integral smooth projective surface Lemma
• Bertini smoothness away from the base locus Theorem
• Every finite Galois extension of an infinite field has a normal basis Theorem
• If deg_xᵢP<| Sᵢ| for each i and P vanishes on S₁×⋯× Sₙ, then P=0 Theorem
• Polynomial algebras over fields have finite integral closures Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Neil Donaldson, Math 120B Notes, Section 22, More general constructions (standard reference, not scraped)