Irreducible and prime elements of an integral domain

Depends on

• Divisibility and associates in an integral domain
• Left inverse, right inverse, and invertible element of a monoid

Used by

• An irreducible polynomial over a field is separable exactly when its derivative is nonzero Corollary
• The reduction of Φₙ is irreducible over F_q exactly when [q] generates (ℤ/n)^× Corollary
• Tangent cone and tangent lines at a point Definition
• The p-primary component of a module over a domain Definition
• Unique factorisation domain Definition
• A nondegenerate projective quadric Example
• A nonreduced equation can hide a smooth hypersurface Example
• A projective cone with a smooth conic base Example
• A regular hyperplane has a one-sheeted projection Example
• An ordinary node has two smooth branches Example
• Principal divisors on the projective line have degree zero Example
• The coordinate axes form a reduced crossing Example
• The cusp y²=x³ has Puiseux parameter (t²,t³) Example
• The plane branch y²=x⁵ has Puiseux parameter (t²,t⁵) Example
• The rank-one 2 by 2 determinantal cone Example
• A reduced prepared hypersurface stays reduced nearby Lemma
• Coprime positive-degree plane forms form a regular sequence Lemma
• Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
• Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
• Every irreducible element of a principal ideal domain is prime Lemma
• Every nonzero nonunit polynomial over a field factors into irreducible polynomials Lemma
• Every prime element of an integral domain is irreducible Lemma
• Finite local length exactly when no common local branch Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• Gauss lemma over a UFD Lemma
• Irreducible holomorphic germs are prime Lemma
• Multiplicity one characterises smooth points with a unique tangent Lemma
• Noetherian domains are atomic Lemma
• The truncated multiplication map is injective exactly when the tangent cones are coprime Lemma
• Total fractions split over the branches of a reduced hypersurface Lemma
• Φ_pʳ(t)=∑_k<pt^kpʳ⁻¹, and Φ_pʳ(t+1) is Eisenstein at p Proposition
• Φₙ is irreducible over K exactly when [K(ζₙ):K]=φ(n), exactly when the embedding into (ℤ/n)^× is onto Proposition
• A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field Theorem
• Every irreducible polynomial over a field is prime Theorem
• Finite unique irreducible components of a hypersurface germ Theorem
• For a nonconstant p in F[x], the ideal (p) is maximal and F[x]/(p) is a field exactly when p is irreducible Theorem
• For gcd(n,q)=1 the reduction of Φₙ in F_q[t] is a product of distinct monic irreducibles, each of degree the order of [q] modulo n Theorem
• Gauss lemma: primitive factorisations over ℚ can be cleared to primitive factorisations over ℤ Theorem
• Symmetry, additivity and local nature of intersection multiplicity Theorem
• Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

…and 1 more result.

Dependency tree · two levels

6 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

• Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)