Factor theorem over a commutative ring

Depends on

• Division by a monic polynomial over a commutative ring
• Evaluation and roots of a polynomial in a commutative target ring
• Universal property of $R[x]$: a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism

Used by

• An irreducible polynomial over ℝ has degree 1 or an even degree Corollary
• Every finite-dimensional endomorphism over an algebraically closed field has Jordan form Corollary
• Every finite-dimensional endomorphism over an algebraically closed field is triangularisable Corollary
• A degree-four polynomial can be reducible over ℚ without having a rational root Counterexample
• Quadratics can have four roots over ℤ/6 and ℤ/8 Counterexample
• Algebraic multiplicity as the exponent of x-λ in χ_T, and geometric multiplicity as dim E_λ(T) Definition
• Repeated roots in extension fields and separable polynomials Definition
• F[x]₍ₓ₎ is the ring of rational functions defined at 0, with maximal ideal generated by x and residue field F Example
• The divisor-sum identity at q=2, n=3 finds exactly two monic irreducible cubics Example
• The four roots of t⁴+t+1 over F₂ are the Frobenius powers of any one of them Example
• 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
• Kronecker's one-root step: adjoining a root removes a linear factor and lowers the remaining degree Lemma
• A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension Proposition
• A base-field isomorphism extends to an isomorphism between splitting fields of corresponding polynomials Theorem
• A monic irreducible of degree d over F_q has the d distinct roots α,α^q,…,α^qᵈ⁻¹ Theorem
• A nonzero polynomial of degree n over an integral domain has at most n distinct roots Theorem
• A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field Theorem
• A root is repeated exactly when it is also a root of the formal derivative Theorem
• An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors Theorem
• Over a field whose characteristic does not divide n, the roots of Φₙ are exactly the primitive roots of unity Theorem

Dependency tree · two levels

11 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, Corollary 17.8 (standard reference, not scraped)
• Neil Donaldson, Math 120B Notes, Theorem 23.14 (standard reference, not scraped)