The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Depends on

• Commutative ring
• A finite sum in a commutative monoid indexed by an arbitrary finite set

Used by

• The generic point of the affine line has no relative k-valued coordinate Counterexample
• The polynomial space admits no complete norm Counterexample
• Content and primitive integer polynomials Definition
• Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree Definition
• Evaluation and roots of a polynomial in a commutative target ring Definition
• Finite linear invariant and coinvariant polynomial algebras Definition
• Finite semisimple Lie algebras and the symmetric adjoint action Definition
• For A∈ Mₙ(F), the characteristic polynomial is χ_A(x)=det(xIₙ-A) when n≥1, with χ_A(x)=1 for the unique 0×0 matrix Definition
• Formal power series over a commutative ring and the coefficient-extraction functional [xⁿ] Definition
• homogeneous polynomial and homogeneous ideal Definition
• Monomials, coefficients, degree in each variable and total degree in F[x₁,…,xₙ] Definition
• Polynomial evaluation at an endomorphism: p(T)=∑ₖ aₖTᵏ Definition
• Polynomial rings in finitely many commuting indeterminates by iteration Definition
• The affine scheme of dual numbers Definition
• The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i Definition
• The cyclotomic polynomials Φₙ∈ℤ[t], defined by ∏_d∣ nΦ_d=tⁿ-1 Definition
• The formal derivative of a polynomial Definition
• A basic open of the affine line Example
• A truncated polynomial local ring is Henselian Example
• For the polynomial space F[x], the canonical map to the algebraic double dual is injective but not surjective Example
• Identifying the coefficient algebra in a concrete Artin–Tate tower Example
• Polynomial addition and multiplication computed from coefficient convolution Example
• S⊗_RR[x]≅ S[x] as S-algebras Example
• Tensoring the injection k[x] →(· x) k[x] with k[x]/(x) gives the zero map Example
• The subalgebra k[x,xy,xy²,…] of k[x,y] is not Noetherian Example
• The subring k[x,y,x/y,x/y²,…] of k(x,y) has a strictly ascending chain of principal ideals Example
• Working the Hilbert basis construction on an ideal of ℤ[x] with non-monic stages Example
• ℤ[x] represents the underlying-set functor on unital rings Example
• False statement: in a Noetherian ring there is a single bound on the number of generators an ideal needs False statement
• A simple residue root determines a coprime residue factorisation Lemma
• A single cancellation step lowers the degree of a polynomial in an ideal once its leading coefficient lies in a realised stage Lemma
• Coefficientwise sums and convolution products of finitely supported sequences are finitely supported Lemma
• For a finite group of ring automorphisms the orbit polynomial is monic over the invariant subring, so the ring is integral over its invariants Lemma
• Gauss lemma over a UFD Lemma
• Lift a Bezout identity for coprime residue factors Lemma
• The leading coefficients of the degree-n elements of an ideal of R[x], together with 0, form an ideal of R, and these ideals ascend with n Lemma
• Degree inequalities for sums and products over a commutative ring Proposition
• Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials Proposition
• In characteristic p the only pᵏ-th root of unity is 1, and t^pᵏ-1=(t-1)^pᵏ Proposition
• Formal polynomials are not the functions they induce Remark

…and 3 more results.

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

• Thomas W. Judson, Abstract Algebra: Theory and Applications, Chapter 17.1 (standard reference, not scraped)
• Neil Donaldson, Math 120B Notes, Section 22 (standard reference, not scraped)