IsMixed — Mathlib

English

If h is IsUnmixed K φ, then IsReal(φ ∘ algebraMap K L) ↔ IsReal φ.

Русский

Если h задаёт IsUnmixed K φ, тогда IsReal(φ ∘ algebraMap K L) ↔ IsReal φ.

LaTeX

$$IsUnmixed K φ → (IsReal (φ ∘ algebraMap K L) ↔ IsReal φ)$$

Lean4

/-- If `L/K` and `φ : L →+* ℂ`, then `IsMixed K φ` if the image of `φ` is complex while the image
of `φ` restricted to `K` is real.

This is the complex embedding analogue of `InfinitePlace.IsRamified K w`, where
`w : InfinitePlace L`. It is not the same concept because conjugation of `φ` in this case
leads to two distinct mixed embeddings but only a single ramified place `w`, leading to a
two-to-one isomorphism between them. -/
abbrev IsMixed (φ : L →+* ℂ) :=
  ComplexEmbedding.IsReal (φ.comp (algebraMap K L)) ∧ ¬ComplexEmbedding.IsReal φ

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