Infinite places of a number field #
This file defines the infinite places of a number field.
Main Definitions and Results #
NumberField.InfinitePlace
: the type of infinite places of a number fieldK
.NumberField.InfinitePlace.mk_eq_iff
: two complex embeddings define the same infinite place iff they are equal or complex conjugates.NumberField.InfinitePlace.IsReal
: The predicate on infinite places saying that a place is real, i.e., defined by a real embedding.NumberField.InfinitePlace.IsComplex
: The predicate on infinite places saying that a place is complex, i.e., defined by a complex embedding that is not real.NumberField.InfinitePlace.mult
: the multiplicity of an infinite place, that is the number of distinct complex embeddings that define it. So it is equal to1
if the place is real and2
if the place is complex.NumberField.InfinitePlace.prod_eq_abs_norm
: the infinite part of the product formula, that is forx ∈ K
, we haveΠ_w ‖x‖_w = |norm(x)|
where the product is over the infinite placew
and‖·‖_w
is the normalized absolute value forw
.NumberField.InfinitePlace.card_add_two_mul_card_eq_rank
: the degree ofK
is equal to the number of real places plus twice the number of complex places.
Tags #
number field, infinite places
An infinite place of a number field K
is a place associated to a complex embedding.
Equations
- NumberField.InfinitePlace K = { w : AbsoluteValue K ℝ // ∃ (φ : K →+* ℂ), NumberField.place φ = w }
Instances For
Return the infinite place defined by a complex embedding φ
.
Equations
Instances For
Equations
- NumberField.InfinitePlace.instFunLikeReal = { coe := fun (w : NumberField.InfinitePlace K) (x : K) => ↑w x, coe_injective' := ⋯ }
For an infinite place w
, return an embedding φ
such that w = infinite_place φ
.
Instances For
An infinite place is real if it is defined by a real embedding.
Equations
- w.IsReal = ∃ (φ : K →+* ℂ), NumberField.ComplexEmbedding.IsReal φ ∧ NumberField.InfinitePlace.mk φ = w
Instances For
An infinite place is complex if it is defined by a complex (ie. not real) embedding.
Equations
- w.IsComplex = ∃ (φ : K →+* ℂ), ¬NumberField.ComplexEmbedding.IsReal φ ∧ NumberField.InfinitePlace.mk φ = w
Instances For
The real embedding associated to a real infinite place.
Equations
Instances For
The multiplicity of an infinite place, that is the number of distinct complex embeddings that
define it, see card_filter_mk_eq
.
Instances For
The map from real embeddings to real infinite places as an equiv
Equations
- NumberField.InfinitePlace.mkReal = Equiv.ofBijective (fun (φ : { φ : K →+* ℂ // NumberField.ComplexEmbedding.IsReal φ }) => ⟨NumberField.InfinitePlace.mk ↑φ, ⋯⟩) ⋯
Instances For
The map from nonreal embeddings to complex infinite places
Instances For
The infinite part of the product formula : for x ∈ K
, we have Π_w ‖x‖_w = |norm(x)|
where
‖·‖_w
is the normalized absolute value for w
.
The number of infinite real places of the number field K
.
Equations
Instances For
Alias of NumberField.InfinitePlace.nrRealPlaces
.
The number of infinite real places of the number field K
.
Instances For
The number of infinite complex places of the number field K
.
Equations
Instances For
Alias of NumberField.InfinitePlace.nrComplexPlaces
.
The number of infinite complex places of the number field K
.
Instances For
The infinite place of the rationals. #
The infinite place of ℚ
, coming from the canonical map ℚ → ℂ
.