Theorems · Theorem · number theory

NumberField.IsCMField.complexConj_eq_self_iff

∀ (K : Type u_1) [inst : Field K] [inst_1 : CharZero K] [inst_2 : NumberField.IsCMField K]
  [inst_3 : Algebra.IsIntegral ℚ K] (x : K),
  (NumberField.IsCMField.complexConj K) x = x ↔ x ∈ NumberField.maximalRealSubfield K

An element of K is fixed by the complex conjugation iff it lies in K⁺.

Defined in: Mathlib.NumberTheory.NumberField.CMField
Cited by: 1 results in Mathlib
Foundations: Depth 307 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field CharZero NumberField.IsCMField Algebra.IsIntegral

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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
Top.topproof · cited by 9,680
Fieldstatement and proof · cited by 7,404
Algebra.algebraMapproof · cited by 4,706
Subgroupproof · cited by 3,593
AlgEquivstatement and proof · cited by 1,681
IntermediateFieldproof · cited by 988
CharZerostatement and proof · cited by 932
Subfieldstatement · cited by 303
Algebra.IsIntegralstatement and proof · cited by 224
NumberField.IsCMFieldstatement and proof · cited by 45
NumberField.maximalRealSubfieldstatement and proof · cited by 38

Cited by1

Results whose statement or proof uses this declaration.

NumberField.IsCMField.RingOfIntegers.complexConj_eq_self_iffproof · cited by 2