Cyclotomic.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

6 declarations, 0 contain sorry, 0 declared as axioms

The cyclotomic character

The cyclotomic character of the absolute Galois group of ℚ, with values in the units of ZHat, together with the auxiliary lemmas needed to construct it.

Imports (3)

Imported by 1 file

FLT.lean

lemma IsAlgClosed.card_rootsOfUnity line 26 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, CharZero, IsAlgClosed, Nat, NeZero, Eq, Nat.card, Subtype, Units, Membership.mem, Subgroup, rootsOfUnity
Proof uses: IsPrimitiveRoot, Exists, IsCyclotomicExtension.exists_isPrimitiveRoot, Top.top, Set, of_eq_true, Set.univ, NeZero.ne, IsPrimitiveRoot.card_rootsOfUnity
35 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma PNat.coe_dvd_iff line 32 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PNat, Iff, Dvd.dvd, Nat
Proof uses: Iff.symm, PNat.dvd_iff
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma rootsOfUnity.pow_eq_pow line 34 source
standard axioms: propext what this means
Statement uses: Nat, Group, Eq, HPow.hPow, OfNat.ofNat, Nat.ModEq
Proof uses: Eq.mpr, HMod.hMod, id, congrArg, pow_eq_pow_mod, congr_arg
22 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma PNat.castHom_val_modEq line 39 source
standard axioms: Quot.sound, propext what this means
Statement uses: Nat, PNat, Dvd.dvd, ZMod, Nat.ModEq, ZMod.val, RingHom, ZMod.castHom
Proof uses: Eq.mpr, ZMod.cast, id, Eq, congrArg, ZMod.castHom_apply, ZMod.cast_eq_val, HMod.hMod, ZMod.val_natCast, Nat.mod_modEq
29 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def CyclotomicCharacterAux The cyclotomic character line 45 source
Source documentation
The cyclotomic character

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, CharZero, IsAlgClosed, MonoidHom, RingEquiv, ZHat
Proof uses: MonoidHom.mk, OneHom.mk, Subtype.mk, PNat, ZMod, Membership.mem, Subring, Subring.mk, Subsemiring.mk, Submonoid.mk, Subsemigroup.mk, Set.ofPred, Dvd.dvd, Nat, Eq, RingHom, ZMod.castHom, Set, Units, modularCyclotomicCharacter
73 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
noncomputable def CyclotomicCharacterAux : (L ≃+* L) →* ZHat where
…
```
Mentions: ZHat
def CyclotomicCharacterZHat The Zhat-adic cyclotomic character line 62 source
Source documentation
The Zhat-adic cyclotomic character.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, CharZero, IsAlgClosed, MonoidHom, RingEquiv, Units, ZHat
Proof uses: MonoidHom.toHomUnits, CyclotomicCharacterAux
33 instances, coercions and auxiliary constants left out.
```
noncomputable def CyclotomicCharacterZHat : (L ≃+* L) →* ZHatˣ :=
…
```