Universal property of $R[x]$: a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism

Depends on

• Evaluation and roots of a polynomial in a commutative target ring
• Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring
• Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials
• Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$
• Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule

Used by

• A polynomial ring on a finite ordered family agrees canonically with the iterated polynomial-ring construction Corollary
• Factor theorem over a commutative ring Corollary
• For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries Corollary
• R[x] is Noetherian if and only if R is Noetherian Corollary
• 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
• Repeated roots in extension fields and separable polynomials Definition
• Subalgebra generated by a subset, algebras of finite type, and module-finite algebras Definition
• F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F Example
• k[x,y]/(xy) and ℤ[x]/(x²-2) are Noetherian without classifying their ideals Example
• The functor of points of the affine line Example
• The minimal polynomial of √2+√3 over ℚ is x⁴-4x²+1 Example
• The parabola y=x² has coordinate ring k[t] and isomorphic intrinsic geometry to the affine line Example
• The polynomial ring in countably many variables is not Noetherian Example
• The symmetric polynomials as the invariant ring of the symmetric group, seen through Noether's finiteness theorem Example
• Translation turns x⁴+1 into an Eisenstein polynomial Example
• Vanishing sets and vanishing ideals form a contravariant Galois connection Example
• A field isomorphism transports polynomials coefficientwise and carries roots, factorizations, and splitting to roots, factorizations, and splitting Lemma
• A finitely localized polynomial ring in positive dimension is not a field 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
• 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
• k-points of k[x₁,..., xₙ]/I are exactly k-algebra maps to k Lemma
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element Lemma
• The monic gcd of two base-field polynomials is unchanged after extending the coefficient field Lemma
• The product of primitive integer polynomials is primitive, and contents multiply Lemma
• Φ₁(0)=-1 and Φₙ(0)=1 for n≥2 Lemma
• Φ_pʳ(t)=∑_k<pt^kpʳ⁻¹, and Φ_pʳ(t+1) is Eisenstein at p Proposition
• A nonzero polynomial over a field is separable exactly when its gcd with its derivative is 1 Theorem
• Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms Theorem
• Eisenstein criterion over the integers Theorem
• Every finite Galois extension of an infinite field has a normal basis Theorem
• Every nonempty principal open is a classical affine variety Theorem
• F[x]/(p) for monic irreducible p is a field extension containing the root x+(p) with unique reduced representatives Theorem
• For every n≥1 there are infinitely many primes p with p≡1 (mod n) Theorem
• Irreducibility after reduction modulo a prime implies irreducibility over ℚ when the leading coefficient survives Theorem
• Polynomial functions on an affine algebraic set are its coordinate ring Theorem
• The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element Theorem
• Universal property of adjoining a root of an irreducible polynomial Theorem

Cited to discharge well-definedness by Evaluation and roots of a polynomial in a commutative target ring.

Dependency tree · two levels

17 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

• James McKernan, MIT 18.703 Lecture 21, Lemma 21.3 (standard reference, not scraped)