For every field $F$, $F[x]$ is a unique factorisation domain

Depends on

• Every nonzero nonunit polynomial over a field factors into irreducible polynomials
• Every irreducible polynomial over a field is prime
• Unique factorisation domain
• A polynomial ring over an integral domain is an integral domain

Used by

• A characteristic polynomial that splits into distinct linear factors forces diagonalisability Corollary
• Every finite family of nonzero polynomials has a splitting field, obtained from their product Corollary
• Every idempotent endomorphism is diagonalisable and is projection onto its image along its kernel Corollary
• The least eventual recurrence order is the degree of the reduced denominator Corollary
• A reduced field spectrum becomes nonreduced Counterexample
• Fₚ(s,t)/Fₚ(sᵖ,tᵖ) has degree p², infinitely many intermediate fields, and no primitive element Counterexample
• Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T) Definition
• Fₚ(t)/Fₚ(tᵖ) is purely inseparable of degree p and separable degree one Example
• Over F₂, x⁴+x²+1=(x²+x+1)² has two distinct roots, each repeated, in its four-element splitting field Example
• xᵖ-t is irreducible and inseparable over Fₚ(t) Example
• A simple finite extension has only finitely many intermediate fields Lemma
• Some vector has vector annihilator equal to the minimal polynomial Lemma
• The rational function field k(t) is not finite over k[t] Lemma
• A nonempty intersection of normal subextensions inside a common algebraic extension is normal Proposition
• If K/F is normal and F⊆ E⊆ K, then K/E is normal Proposition
• A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials Theorem
• A complex polynomial of degree n has exactly n roots counted with multiplicity Theorem
• Eisenstein criterion over the integers 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
• If χ_T(x)=∏_i<n(x-λᵢ) in F[x], then χ_p(T)(y)=∏_i<n(y-p(λᵢ)) for every p∈ F[x]: the eigenvalues of p(T) are p(λᵢ), counted with algebraic multiplicity Theorem
• Primary decomposition: the irreducible-power factors of μ_T split V into their invariant kernels Theorem
• The restriction of a diagonalisable endomorphism to an invariant subspace is diagonalisable Theorem
• The ring of holomorphic germs is a UFD Theorem

Dependency tree · two levels

12 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

• Neil Donaldson, Math 120B Notes, Theorem 23.11 (standard reference, not scraped)