Theorems · Theorem · number theory

NumberField.Units.regOfFamily_eq_det

∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K]
  (u : Fin (NumberField.Units.rank K) → (NumberField.RingOfIntegers K)ˣ) (w' : NumberField.InfinitePlace K)
  (e : { w // w ≠ w' } ≃ Fin (NumberField.Units.rank K)),
  NumberField.Units.regOfFamily u =
    |(Matrix.of fun i w => ↑(↑w).mult * Real.log (↑w ((algebraMap (NumberField.RingOfIntegers K) K) ↑(u (e i))))).det|

For any infinite place w', the regulator of the family u is equal to the absolute value of the determinant of the matrix with entries (mult w * log w (u i))_i for w ≠ w'.

Defined in: Mathlib.NumberTheory.NumberField.Units.Regulator
Cited by: 3 results in Mathlib
Foundations: Depth 308 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field NumberField

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Cites24

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Realstatement · cited by 25,697
RingHomstatement · cited by 10,189
Equivstatement and proof · cited by 8,337
Fieldstatement and proof · cited by 7,404
Algebra.algebraMapstatement and proof · cited by 4,706
Matrixstatement · cited by 4,303
Equiv.symmproof · cited by 3,681
Unitsstatement and proof · cited by 2,804
Units.valstatement and proof · cited by 1,966
absstatement and proof · cited by 1,814
Real.logstatement and proof · cited by 939

Cited by3

Results whose statement or proof uses this declaration.

NumberField.Units.finrank_mul_regOfFamily_eq_detproof · cited by 2
NumberField.IsCMField.regOfFamily_realFunSystemproof · cited by 1
NumberField.Units.regulator_eq_detproof · cited by 0