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

noncomputable def Algebra.norm (R : Type u_1) {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] : S →* R

The norm of an element s of an R-algebra is the determinant of (*) s.

Equations

• Algebra.norm R = LinearMap.det.comp ↑(Algebra.lmul R S).toRingHom

theorem Algebra.norm_apply (R : Type u_1) {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] (x : S) : (Algebra.norm R) x = LinearMap.det ((Algebra.lmul R S) x)

theorem Algebra.norm_eq_one_of_not_exists_basis (R : Type u_1) {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] (h : ¬∃ (s : Finset S), Nonempty (Basis { x : S // x ∈ s } R S)) (x : S) : (Algebra.norm R) x = 1

theorem Algebra.norm_eq_one_of_not_module_finite {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] (h : ¬Module.Finite R S) (x : S) : (Algebra.norm R) x = 1

theorem Algebra.norm_eq_matrix_det {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] {ι : Type w} [Fintype ι] [DecidableEq ι] (b : Basis ι R S) (s : S) : (Algebra.norm R) s = ((Algebra.leftMulMatrix b) s).det

theorem Algebra.norm_algebraMap_of_basis {R : Type u_1} {S : Type u_2} [CommRing R] [Ring S] [Algebra R S] {ι : Type w} [Fintype ι] (b : Basis ι R S) (x : R) : (Algebra.norm R) ((algebraMap R S) x) = x ^ Fintype.card ι

If x is in the base ring K, then the norm is x ^ [L : K].

@[simp]

theorem Algebra.norm_algebraMap {K : Type u_3} [Field K] {L : Type u_4} [Ring L] [Algebra K L] (x : K) : (Algebra.norm K) ((algebraMap K L) x) = x ^ Module.finrank K L

If x is in the base field K, then the norm is x ^ [L : K].

(If L is not finite-dimensional over K, then norm = 1 = x ^ 0 = x ^ (finrank L K).)