Reindexed basis

This file introduces an equivalence between the set of embeddings of K into ℂ and the index set of the chosen basis of the ring of integers of K.

Tagshouse

number field, algebraic number

@[reducible, inline]

abbrev NumberField.equivReindex (K : Type u_1) [Field K] [NumberField K] : (K →+* ℂ) ≃ Module.Free.ChooseBasisIndex ℤ (NumberField.RingOfIntegers K)

An equivalence between the set of embeddings of K into ℂ and the index set of the chosen basis of the ring of integers of K.

Equations

• NumberField.equivReindex K = Fintype.equivOfCardEq ⋯

@[reducible, inline]

abbrev NumberField.basisMatrix (K : Type u_1) [Field K] [NumberField K] : Matrix (K →+* ℂ) (K →+* ℂ) ℂ

The basis matrix for the embeddings of K into ℂ. This matrix is formed by taking the lattice basis vectors of K and reindexing them according to the equivalence equivReindex, then transposing the resulting matrix.

Equations

• NumberField.basisMatrix K = Matrix.of fun (i : K →+* ℂ) => (NumberField.canonicalEmbedding.latticeBasis K) ((NumberField.equivReindex K) i)

theorem NumberField.det_of_basisMatrix_non_zero (K : Type u_1) [Field K] [NumberField K] [DecidableEq (K →+* ℂ)] : (NumberField.basisMatrix K).det ≠ 0

instance NumberField.instInvertibleMatrixRingHomComplexBasisMatrix (K : Type u_1) [Field K] [NumberField K] [DecidableEq (K →+* ℂ)] : Invertible (NumberField.basisMatrix K)

Equations

• NumberField.instInvertibleMatrixRingHomComplexBasisMatrix K = (NumberField.basisMatrix K).invertibleOfIsUnitDet ⋯

theorem NumberField.canonicalEmbedding_eq_basisMatrix_mulVec {K : Type u_1} [Field K] [NumberField K] (α : K) : (NumberField.canonicalEmbedding K) α = (NumberField.basisMatrix K).transpose.mulVec fun (i : K →+* ℂ) => ↑((((NumberField.integralBasis K).reindex (NumberField.equivReindex K).symm).repr α) i)

theorem NumberField.inverse_basisMatrix_mulVec_eq_repr {K : Type u_1} [Field K] [NumberField K] [DecidableEq (K →+* ℂ)] (α : NumberField.RingOfIntegers K) (i : K →+* ℂ) : (NumberField.basisMatrix K).transpose⁻¹.mulVec (fun (j : K →+* ℂ) => (NumberField.canonicalEmbedding K) ((algebraMap (NumberField.RingOfIntegers K) K) α) j) i = ↑((((NumberField.integralBasis K).reindex (NumberField.equivReindex K).symm).repr ↑α) i)