Theorems · Definition · number theory

NumberField.IsCMField.ringOfIntegersComplexConj

(K : Type u_1) →
  [inst : Field K] →
    [inst_1 : CharZero K] →
      [NumberField.IsCMField K] →
        [Algebra.IsIntegral ℚ K] →
          NumberField.RingOfIntegers K ≃ₐ[NumberField.RingOfIntegers ↥(NumberField.maximalRealSubfield K)]
            NumberField.RingOfIntegers K

A variant of the complex conjugation defined as an AlgEquiv on the ring of integers.

Defined in: Mathlib.NumberTheory.NumberField.CMField
Cited by: 3 results in Mathlib
Foundations: Depth 303 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field CharZero NumberField.IsCMField Algebra.IsIntegral

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites10

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

Fieldstatement and proof · cited by 7,404
AlgEquivstatement · cited by 1,681
CharZerostatement and proof · cited by 932
NumberField.RingOfIntegersstatement · cited by 413
Subfieldstatement · cited by 303
Algebra.IsIntegralstatement and proof · cited by 224
NumberField.IsCMFieldstatement and proof · cited by 45
NumberField.maximalRealSubfieldstatement · cited by 38
NumberField.IsCMField.complexConjproof · cited by 14
NumberField.RingOfIntegers.mapAlgEquivproof · cited by 0

Cited by3

Results whose statement or proof uses this declaration.

NumberField.IsCMField.coe_ringOfIntegersComplexConjstatement · cited by 1
NumberField.IsCMField.ringOfIntegersComplexConj.congr_simpstatement and proof · cited by 0
NumberField.IsCMField.ringOfIntegersComplexConj_eq_self_iffstatement and proof · cited by 0