Quadratic.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 10 curated AI-written annotations and 7190 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, Formal Conjectures paper, arXiv v1

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

instance QuadraticAlgebra.instIsQuadraticExtension Any QuadraticAlgebra ℚ a b that is a field is automatically a quadratic extension of ℚ, i.e., a degree-2 extension line 29 source
Source documentation
Any QuadraticAlgebra ℚ a b that is a field is automatically a quadratic extension of ℚ, i.e., a degree-2 extension. Combined with IsQuadraticField.instNumberField, this gives NumberField (QuadraticAlgebra ℚ a b) for free.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Rat, Fact, Ne, HPow.hPow, Nat, OfNat.ofNat, HAdd.hAdd, HMul.hMul, Algebra.IsQuadraticExtension, QuadraticAlgebra
Proof uses: Algebra.IsQuadraticExtension.mk, QuadraticAlgebra.finrank_eq_two
49 instances, coercions and auxiliary constants left out.
```
instance QuadraticAlgebra.instIsQuadraticExtension (a b : ℚ) [Fact (∀ r : ℚ, r ^ 2 ≠ a + b * r)] :
    IsQuadraticExtension ℚ (QuadraticAlgebra ℚ a b) where
…
```
instance Algebra.IsQuadraticExtension.to_numberField A quadratic field is a number field: it has characteristic zero and is finite-dimensional over ℚ line 35 source
Source documentation
A quadratic field is a number field: it has characteristic zero and is finite-dimensional over ℚ.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, CharZero, Algebra.IsQuadraticExtension, Rat, NumberField
Proof uses: NumberField.mk, FiniteDimensional.of_finrank_pos
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
instance Algebra.IsQuadraticExtension.to_numberField (K : Type*) [Field K] [CharZero K]
    [IsQuadraticExtension ℚ K] : NumberField K where
…
```