A polynomial ring over an integral domain is an integral domain

Depends on

• Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring
• Over an integral domain, degrees add under multiplication of nonzero polynomials
• Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors

Used by

• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• A polynomial ring in finitely many indeterminates over an integral domain is an integral domain Corollary
• For a field F, F(t)=Frac(F[t]) is its rational function field; in particular ℝ(t)=Frac(ℝ[t]) Corollary
• For every field F, F[x] is a Euclidean domain with degree as Euclidean function Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• A nonclosed open immersion is not proper Counterexample
• Global generation does not imply very ampleness Counterexample
• Global sections do not determine a sheaf on P1 Counterexample
• Quasi-finite does not imply finite Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The affine line is not proper Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• The degenerate affine Hecke algebra Definition
• The Laurent polynomial ring as the principal localisation of Z[t] at t Definition
• A closed point immersion is unramified Example
• Normalization of a nodal affine plane curve Example
• Normalization of the cusp semigroup ring Example
• Normalization of the t³,t⁴,t⁵ monomial curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• Tensoring the injection k[x] →(· x) k[x] with k[x]/(x) gives the zero map Example
• The punctured affine line as an open finite factorization Example
• The quotient of GL2 by the diagonal torus is the complement of the diagonal in P1 x P1 Example
• The tangent space of the parabola at a general point and the local parameter Example
• Φ₁ through Φ₁₂ computed from the divisor recursion Example
• FALSE: every finitely generated module over a domain is a direct sum of cyclic modules False statement
• A quasi-finite one-generator quotient is locally its integral closure Lemma
• A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings Lemma
• Coprime positive-degree plane forms form a regular sequence Lemma
• Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
• Finite algebras over a strongly transcendental variable are nowhere quasi-finite Lemma
• Finite local length exactly when no common local branch Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• Fraction-field coefficients of an integral LKB class are Laurent polynomials Lemma
• If p is a prime not dividing n, a rational minimal polynomial of a primitive n-th root of unity also kills its p-th power Lemma
• Polynomial rings over normal domains are normal Lemma
• Prime and local-ring correspondence on standard projective charts Lemma
• The eventual Hilbert function of a zero-dimensional projective quotient equals its total length Lemma
• The fork-noodle pairing is well defined and equivariant Lemma
• The product of primitive integer polynomials is primitive, and contents multiply Lemma

…and 9 more results.

Dependency tree · two levels

11 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

• Thomas W. Judson, Abstract Algebra: Theory and Applications, Theorem 17.4 (standard reference, not scraped)