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Every nonzero nonunit polynomial over a field factors into irreducible polynomials
Statement
Every nonzero nonunit polynomial over a field is a finite product of irreducible polynomials.
Facts & Assumptions
Given: A field and a nonzero nonunit polynomial .
A nonzero nonunit is irreducible when every factorization has a unit factor (Irreducible and prime elements of an integral domain).
Degrees add under multiplication of nonzero polynomials over a field (Over an integral domain, degrees add under multiplication of nonzero polynomials).
The units of are exactly its nonzero constant polynomials (The units of over an integral domain are exactly the constant polynomials whose values are units of ).
Strong induction proves a natural-number property once the case at follows from all smaller cases (Strong (complete) induction).
Proof
Use strong induction on ; by [L3], a nonzero nonunit has .
If is irreducible, it is already a one-factor product; otherwise [L1] gives with nonunits, and neither is zero because .
By [L2], and are positive and strictly below , so the induction hypotheses factor both into irreducibles; concatenating those factorizations gives one for , and [L4] completes the induction.
Depends on
Used by
- Every nonconstant polynomial over a field has a root in some field extension Corollary
- Every real polynomial factors into linear and irreducible quadratic factors Corollary
- A nonzero polynomial over a field is separable exactly when its gcd with its derivative is 1 Theorem
- For every field F, F[x] is a unique factorisation domain Theorem
- In characteristic p, every irreducible polynomial is uniquely g(x^pᵉ) with g irreducible and separable Theorem
- Over F_q, x^qⁿ-x is the product of all monic irreducibles whose degrees divide n Theorem
Dependency tree · two levels
13 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 23.7 (standard reference, not scraped)