Norm for (finite) ring extensions

Suppose we have an R-algebra S with a finite basis. For each s : S, the determinant of the linear map given by multiplying by s gives information about the roots of the minimal polynomial of s over R.

Implementation notes

Typically, the norm is defined specifically for finite field extensions. The current definition is as general as possible and the assumption that we have fields or that the extension is finite is added to the lemmas as needed.

We only define the norm for left multiplication (Algebra.leftMulMatrix, i.e. LinearMap.mulLeft). For now, the definitions assume S is commutative, so the choice doesn't matter anyway.

See also Algebra.trace, which is defined similarly as the trace of Algebra.leftMulMatrix.

References

• https://en.wikipedia.org/wiki/Field_norm

theorem Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] (pb : PowerBasis R S) : (Algebra.norm R) pb.gen = (-1) ^ pb.dim * (minpoly R pb.gen).coeff 0

Given pb : PowerBasis K S, then the norm of pb.gen is (-1) ^ pb.dim * coeff (minpoly K pb.gen) 0.

theorem Algebra.PowerBasis.norm_gen_eq_prod_roots {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] {F : Type u_6} [Field F] [Algebra R F] (pb : PowerBasis R S) (hf : Polynomial.Splits (algebraMap R F) (minpoly R pb.gen)) : (algebraMap R F) ((Algebra.norm R) pb.gen) = ((minpoly R pb.gen).aroots F).prod

Given pb : PowerBasis R S, then the norm of pb.gen is ((minpoly R pb.gen).aroots F).prod.

@[simp]

theorem Algebra.norm_zero {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] [Nontrivial S] [Module.Free R S] [Module.Finite R S] : (Algebra.norm R) 0 = 0

@[simp]

theorem Algebra.norm_eq_zero_iff {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {x : S} : (Algebra.norm R) x = 0 ↔ x = 0

theorem Algebra.norm_ne_zero_iff {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {x : S} : (Algebra.norm R) x ≠ 0 ↔ x ≠ 0

@[simp]

theorem Algebra.norm_eq_zero_iff' {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S] {x : S} : LinearMap.det ((LinearMap.mul R S) x) = 0 ↔ x = 0

This is Algebra.norm_eq_zero_iff composed with Algebra.norm_apply.

theorem Algebra.norm_eq_zero_iff_of_basis {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] {ι : Type w} [Finite ι] [IsDomain R] [IsDomain S] (b : Basis ι R S) {x : S} : (Algebra.norm R) x = 0 ↔ x = 0

theorem Algebra.norm_ne_zero_iff_of_basis {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] {ι : Type w} [Finite ι] [IsDomain R] [IsDomain S] (b : Basis ι R S) {x : S} : (Algebra.norm R) x ≠ 0 ↔ x ≠ 0

theorem Algebra.norm_eq_norm_adjoin (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] (x : L) : (Algebra.norm K) x = (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K x) ^ Module.finrank (↥K⟮x⟯) L

theorem IntermediateField.AdjoinSimple.norm_gen_eq_one {K : Type u_4} {L : Type u_5} [Field K] [Field L] [Algebra K L] {x : L} (hx : ¬IsIntegral K x) : (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K x) = 1

theorem IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots {K : Type u_4} {L : Type u_5} {F : Type u_6} [Field K] [Field L] [Field F] [Algebra K L] [Algebra K F] (x : L) (hf : Polynomial.Splits (algebraMap K F) (minpoly K x)) : (algebraMap K F) ((Algebra.norm K) (IntermediateField.AdjoinSimple.gen K x)) = ((minpoly K x).aroots F).prod

theorem Algebra.norm_eq_prod_embeddings_gen {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] (F : Type u_6) [Field F] [Algebra R F] (pb : PowerBasis R S) (hE : Polynomial.Splits (algebraMap R F) (minpoly R pb.gen)) (hfx : IsSeparable R pb.gen) : (algebraMap R F) ((Algebra.norm R) pb.gen) = ∏ σ : S →ₐ[R] F, σ pb.gen

theorem Algebra.norm_eq_prod_roots {K : Type u_4} {L : Type u_5} (F : Type u_6) [Field K] [Field L] [Field F] [Algebra K L] [Algebra K F] [Algebra.IsSeparable K L] [FiniteDimensional K L] {x : L} (hF : Polynomial.Splits (algebraMap K F) (minpoly K x)) : (algebraMap K F) ((Algebra.norm K) x) = ((minpoly K x).aroots F).prod ^ Module.finrank (↥K⟮x⟯) L

theorem Algebra.prod_embeddings_eq_finrank_pow {K : Type u_4} {L : Type u_5} (F : Type u_6) [Field K] [Field L] [Field F] [Algebra K L] [Algebra K F] (E : Type u_7) [Field E] [Algebra K E] [Algebra L F] [IsScalarTower K L F] [IsAlgClosed E] [Algebra.IsSeparable K F] [FiniteDimensional K F] (pb : PowerBasis K L) : ∏ σ : F →ₐ[K] E, σ ((algebraMap L F) pb.gen) = (∏ σ : L →ₐ[K] E, σ pb.gen) ^ Module.finrank L F

theorem Algebra.norm_eq_prod_embeddings (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] (E : Type u_7) [Field E] [Algebra K E] [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsAlgClosed E] (x : L) : (algebraMap K E) ((Algebra.norm K) x) = ∏ σ : L →ₐ[K] E, σ x

For L/K a finite separable extension of fields and E an algebraically closed extension of K, the norm (down to K) of an element x of L is equal to the product of the images of x over all the K-embeddings σ of L into E.

theorem Algebra.norm_eq_prod_automorphisms (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [IsGalois K L] (x : L) : (algebraMap K L) ((Algebra.norm K) x) = ∏ σ : L ≃ₐ[K] L, σ x

theorem Algebra.isIntegral_norm {R : Type u_1} [CommRing R] (K : Type u_4) {L : Type u_5} [Field K] [Field L] [Algebra K L] [Algebra R L] [Algebra R K] [IsScalarTower R K L] [Algebra.IsSeparable K L] [FiniteDimensional K L] {x : L} (hx : IsIntegral R x) : IsIntegral R ((Algebra.norm K) x)

theorem Algebra.norm_eq_of_algEquiv {R : Type u_1} {S : Type u_2} {T : Type u_3} [CommRing R] [Ring S] [Algebra R S] [Ring T] [Algebra R T] (e : S ≃ₐ[R] T) (x : S) : (Algebra.norm R) (e x) = (Algebra.norm R) x

theorem Algebra.norm_eq_of_ringEquiv {A : Type u_8} {B : Type u_9} {C : Type u_10} [CommRing A] [CommRing B] [Ring C] [Algebra A C] [Algebra B C] (e : A ≃+* B) (he : (algebraMap B C).comp ↑e = algebraMap A C) (x : C) : e ((Algebra.norm A) x) = (Algebra.norm B) x

theorem Algebra.norm_eq_of_equiv_equiv {A₁ : Type u_8} {B₁ : Type u_9} {A₂ : Type u_10} {B₂ : Type u_11} [CommRing A₁] [Ring B₁] [CommRing A₂] [Ring B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂) (he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)) (x : B₁) : (Algebra.norm A₁) x = e₁.symm ((Algebra.norm A₂) (e₂ x))

theorem Algebra.norm_norm (K : Type u_4) (L : Type u_5) {F : Type u_6} [Field K] [Field L] [Field F] [Algebra K L] [Algebra K F] [Algebra L F] [IsScalarTower K L F] [Algebra.IsSeparable K F] (x : F) : (Algebra.norm K) ((Algebra.norm L) x) = (Algebra.norm K) x

Let $F / L / K$ be a tower of finite extensions of fields. Then $\text{Norm}_{F/K} = \text{Norm}_{L/K} \circ \text{Norm}_{F/L}$.

Stacks Tag 0BIJ (This is a special case of 0BIJ norm, we require separability here. )