Quotient.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

Norm and the residue field

Material for a new Mathlib.RingTheory.Norm.Quotient (the norm analogue of Mathlib.RingTheory.Trace.Quotient): the norm commutes with reduction to the residue field.

Imports (5)

Imported by 2 files

theorem sq_sub_trace_mul_self_add_norm Rank-2 Cayley–Hamilton line 28 source
Source documentation
Rank-2 Cayley–Hamilton. Every element θ of a free rank-2 A-algebra B satisfies its characteristic polynomial X² - (trace θ) X + (norm θ): this is Cayley–Hamilton (Algebra.aeval_self_charpoly_lmul) applied to left multiplication by θ, whose characteristic polynomial is X² - trace X + norm in rank two (Matrix.charpoly_fin_two).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Nontrivial, Algebra, Module.Free, Module.Finite, Eq, Nat, Module.finrank, OfNat.ofNat, HAdd.hAdd, HSub.hSub, HPow.hPow, HMul.hMul, RingHom, Algebra.algebraMap, LinearMap, RingHom.id, Algebra.trace, MonoidHom, Algebra.norm
Proof uses: Module.Basis, Fin, Module.finBasisOfFinrankEq, rfl, Polynomial, LinearMap.charpoly, AlgHom, Module.End, Algebra.lmul, Polynomial.X, Polynomial.C, Eq.mpr, Matrix.charpoly, LinearEquiv, Matrix, LinearMap.toMatrix, id, congrArg, Eq.symm, LinearMap.charpoly_toMatrix and 22 more
122 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem sq_sub_trace_mul_self_add_norm {A B : Type*} [CommRing A] [Nontrivial A] [CommRing B]
    [Algebra A B] [Module.Free A B] [Module.Finite A B] (h2 : Module.finrank A B = 2) (θ : B) :
    θ ^ 2 - algebraMap A B (Algebra.trace A B θ) * θ + algebraMap A B (Algebra.norm A θ) = 0 := by
…
```
theorem norm_quotient_mk Norm and the residue field line 48 source
Source documentation
Norm and the residue field. For a finite free algebra B over a local ring A, the norm of the reduction of x is the reduction of the norm. This is the norm analogue of Algebra.trace_quotient_mk; the proof is identical, using RingHom.map_det in place of the matrix trace map.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, Module.Free, Module.Finite, Eq, HasQuotient.Quotient, Ideal, IsLocalRing.maximalIdeal, MonoidHom, Ideal.map, RingHom, Algebra.algebraMap, Algebra.norm, Ideal.Quotient.mk
Proof uses: Field, Ideal.Quotient.field, Module.Basis, Module.Free.ChooseBasisIndex, Module.Free.chooseBasis, Eq.mpr, Matrix.det, AlgHom, Matrix, Algebra.leftMulMatrix, IsLocalRing.basisQuotient, id, congrArg, Algebra.norm_eq_matrix_det, RingHom.mapMatrix, RingHom.map_det, DecidableEq, Fintype, Eq.rec, Eq.refl and 22 more
85 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem norm_quotient_mk {A B : Type*} [CommRing A] [CommRing B] [Algebra A B] [IsLocalRing A]
    [Module.Free A B] [Module.Finite A B] (x : B) :
    Algebra.norm (A ⧸ maximalIdeal A)
        (Ideal.Quotient.mk (Ideal.map (algebraMap A B) (maximalIdeal A)) x)
      = Ideal.Quotient.mk (maximalIdeal A) (Algebra.norm A x) := by
…
```
theorem sq_sub_trace_mul_self_add_norm_residue Documented declaration: sq_sub_trace_mul_self_add_norm_residue line 71 source
Source documentation
Transport of the rank-2 Cayley–Hamilton identity θ² - t·θ + n = 0 (t, n the trace and norm of θ, sq_sub_trace_mul_self_add_norm) through an isomorphism of residue fields: the image of the residue of θ satisfies the corresponding relation in the residues of t and n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, Algebra, IsLocalHom, RingHom, Algebra.algebraMap, Module.Free, Module.Finite, Eq, Nat, Module.finrank, OfNat.ofNat, IsLocalRing.ResidueField, AlgEquiv, HAdd.hAdd, HSub.hSub, HPow.hPow, IsLocalRing.residue, HMul.hMul, LinearMap and 4 more
Proof uses: of_eq_true, eq_self, Eq.mp, congr, congrArg, Eq.trans, map_add, map_sub, map_pow, map_mul, congrFun', Eq.symm, outParam, MonoidHom.id, AlgEquiv.commutes, map_zero, sq_sub_trace_mul_self_add_norm
118 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem sq_sub_trace_mul_self_add_norm_residue {S : Type u} [CommRing S] [IsLocalRing S]
    [Algebra R S] [IsLocalHom (algebraMap R S)] [Module.Free R S] [Module.Finite R S]
    (hSrank : Module.finrank R S = 2) {k' : Type u} [CommRing k']
    [Algebra (ResidueField R) k'] (resIso : ResidueField S ≃ₐ[ResidueField R] k') (θ : S) :
    resIso (residue S θ) ^ 2
      - algebraMap (ResidueField R) k' (residue R (Algebra.trace R S θ)) * resIso (residue S θ)
      + algebraMap (ResidueField R) k' (residue R (Algebra.norm R θ)) = 0 := by
…
```