Theorems · Theorem · number theory
NumberField.IsCMField.complexConj_torsion
∀ (K : Type u_1) [inst : Field K] [inst_1 : CharZero K] [inst_2 : NumberField.IsCMField K] [inst_3 : NumberField K]
(ζ : ↥(NumberField.Units.torsion K)),
(NumberField.IsCMField.complexConj K) ((algebraMap (NumberField.RingOfIntegers K) K) ↑↑ζ) =
((algebraMap (NumberField.RingOfIntegers K) K) ↑↑ζ)⁻¹The image of a root of unity by the complex conjugation is its inverse.
This is the version of Complex.conj_rootsOfUnity for CM-fields.
- Defined in
- Mathlib.NumberTheory.NumberField.CMField
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 305 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
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
- RingHomstatement and proof · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Complexproof · cited by 5,565
- Algebra.algebraMapstatement and proof · cited by 4,706
- Subgroupstatement · cited by 3,593
- Unitsstatement and proof · cited by 2,804
- Units.valstatement and proof · cited by 1,966
- AlgEquivstatement · cited by 1,681
- CharZerostatement and proof · cited by 932
- starRingEndproof · cited by 671
- NumberFieldstatement and proof · cited by 653
Cited by1
Results whose statement or proof uses this declaration.
- NumberField.IsCMField.unitsComplexConj_torsionproof · cited by 1