Polynomial rings in finitely many commuting indeterminates by iteration

Depends on

• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
• Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring

Used by

• A polynomial ring in finitely many indeterminates over an integral domain is an integral domain Corollary
• A polynomial ring on a finite ordered family agrees canonically with the iterated polynomial-ring construction Corollary
• For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries Corollary
• If R is Noetherian then R[x₁,…,xₙ] is Noetherian for every n∈ℕ Corollary
• For a field F, the ideal (x,y) in F[x,y] is not principal Counterexample
• Arithmetization of Boolean formulas Definition
• Classical affine algebraic sets, including the empty boundaries Definition
• Lexicographic order on exponent tuples and the multidegree of a nonzero polynomial Definition
• Monomials, coefficients, degree in each variable and total degree in F[x₁,…,xₙ] Definition
• Multilinear extension of a Boolean-cube table Definition
• Multilinear polynomials and the reduction xᵢ²↦ xᵢ on the cube Definition
• Polynomial identity testing Definition
• Subalgebra generated by a subset, algebras of finite type, and module-finite algebras Definition
• Sum-check with explicit degree bounds Definition
• Symmetric polynomials as the invariants of variable permutations Definition
• The cycle index of a finite permutation group Definition
• The elementary symmetric polynomials e₀,e₁,…,eₙ Definition
• The Vandermonde polynomial Δₙ=∏_i<j(xᵢ-xⱼ) Definition
• k[x,y]/(xy) and ℤ[x]/(x²-2) are Noetherian without classifying their ideals Example
• The polynomial ring in countably many variables is not Noetherian 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
• The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem Example
• Vanishing sets and vanishing ideals form a contravariant Galois connection Example
• False statement: in a Noetherian ring there is a single bound on the number of generators an ideal needs False statement
• Tor one of R modulo I and M is not always the I-torsion submodule of M False statement
• A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial Lemma
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element Lemma
• Reducing f modulo gᵢ(xᵢ)=∏_s∈ Sᵢ(xᵢ-s) lowers each deg_xᵢ below | Sᵢ|, preserves the values on the grid, and preserves any top-degree coefficient whose exponents stay below the grid sizes Lemma
• 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
• The elementary symmetric polynomials are algebraically independent over the coefficient ring Theorem
• The Schwartz-Zippel lemma Theorem

Dependency tree · two levels

7 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, More general constructions (standard reference, not scraped)