Splits.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 8 curated AI-written annotations and 1204 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned FLT blueprint introduction

1 declarations, 0 contain sorry, 0 declared as axioms

Polynomials of degree at most two split once they have a root

Material for Mathlib.Algebra.Polynomial.Splits.

Imports (1)

Mathlib.Algebra.Polynomial.Splits

Imported by 2 files

theorem Polynomial.Splits.of_natDegree_le_two_of_isRoot A polynomial of degree at most 2 over a field splits as soon as it has a root: the root splits off a linear factor and the cofactor is linear line 22 source
Source documentation
A polynomial of degree at most 2 over a field splits as soon as it has a root: the root splits off a linear factor and the cofactor is linear.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Polynomial, LE.le, Nat, Polynomial.natDegree, OfNat.ofNat, Polynomial.IsRoot, Polynomial.Splits
Proof uses: Eq, Ne, Or, eq_or_ne, Polynomial.Splits.zero, Eq.symm, HMul.hMul, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, Dvd.dvd, Iff.mpr, Polynomial.dvd_iff_isRoot, Eq.mpr, id, congrArg, MulZeroClass.mul_zero, Eq.refl, Eq.mp and 7 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
```
theorem Splits.of_natDegree_le_two_of_isRoot {k : Type*} [Field k] {p : k[X]}
    (hdeg : p.natDegree ≤ 2) {x : k} (hx : p.IsRoot x) : p.Splits := by
…
```