Documentation

Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Infinite places of a number field #

This file defines the infinite places of a number field.

Main Definitions and Results #

Tags #

number field, infinite places

def NumberField.InfinitePlace (K : Type u_2) [Field K] :
Type u_2

An infinite place of a number field K is a place associated to a complex embedding.

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    noncomputable def NumberField.InfinitePlace.mk {K : Type u_2} [Field K] (φ : K →+* ) :

    Return the infinite place defined by a complex embedding φ.

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      theorem NumberField.InfinitePlace.coe_apply {K : Type u_4} [Field K] (v : InfinitePlace K) (x : K) :
      v x = v x
      theorem NumberField.InfinitePlace.ext {K : Type u_4} [Field K] (v₁ v₂ : InfinitePlace K) (h : ∀ (k : K), v₁ k = v₂ k) :
      v₁ = v₂
      theorem NumberField.InfinitePlace.ext_iff {K : Type u_4} [Field K] {v₁ v₂ : InfinitePlace K} :
      v₁ = v₂ ∀ (k : K), v₁ k = v₂ k
      @[simp]
      theorem NumberField.InfinitePlace.apply {K : Type u_2} [Field K] (φ : K →+* ) (x : K) :
      (mk φ) x = φ x
      noncomputable def NumberField.InfinitePlace.embedding {K : Type u_2} [Field K] (w : InfinitePlace K) :

      For an infinite place w, return an embedding φ such that w = infinite_place φ .

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        theorem NumberField.InfinitePlace.eq_iff_eq {K : Type u_2} [Field K] (x : K) (r : ) :
        (∀ (w : InfinitePlace K), w x = r) ∀ (φ : K →+* ), φ x = r
        theorem NumberField.InfinitePlace.le_iff_le {K : Type u_2} [Field K] (x : K) (r : ) :
        (∀ (w : InfinitePlace K), w x r) ∀ (φ : K →+* ), φ x r
        theorem NumberField.InfinitePlace.pos_iff {K : Type u_2} [Field K] {w : InfinitePlace K} {x : K} :
        0 < w x x 0
        @[simp]
        theorem NumberField.InfinitePlace.mk_eq_iff {K : Type u_2} [Field K] {φ ψ : K →+* } :

        An infinite place is real if it is defined by a real embedding.

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          An infinite place is complex if it is defined by a complex (ie. not real) embedding.

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            theorem NumberField.InfinitePlace.ne_of_isReal_isComplex {K : Type u_2} [Field K] {w w' : InfinitePlace K} (h : w.IsReal) (h' : w'.IsComplex) :
            w w'
            noncomputable def NumberField.InfinitePlace.embedding_of_isReal {K : Type u_2} [Field K] {w : InfinitePlace K} (hw : w.IsReal) :

            The real embedding associated to a real infinite place.

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              @[simp]
              noncomputable def NumberField.InfinitePlace.mult {K : Type u_2} [Field K] (w : InfinitePlace K) :

              The multiplicity of an infinite place, that is the number of distinct complex embeddings that define it, see card_filter_mk_eq.

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                @[simp]
                theorem NumberField.InfinitePlace.mult_isReal {K : Type u_2} [Field K] (w : { w : InfinitePlace K // w.IsReal }) :
                (↑w).mult = 1
                @[simp]
                theorem NumberField.InfinitePlace.mult_isComplex {K : Type u_2} [Field K] (w : { w : InfinitePlace K // w.IsComplex }) :
                (↑w).mult = 2
                theorem NumberField.InfinitePlace.prod_eq_prod_mul_prod {K : Type u_2} [Field K] {α : Type u_4} [CommMonoid α] [NumberField K] (f : InfinitePlace Kα) :
                w : InfinitePlace K, f w = (∏ w : { w : InfinitePlace K // w.IsReal }, f w) * w : { w : InfinitePlace K // w.IsComplex }, f w
                theorem NumberField.InfinitePlace.sum_eq_sum_add_sum {K : Type u_2} [Field K] {α : Type u_4} [AddCommMonoid α] [NumberField K] (f : InfinitePlace Kα) :
                w : InfinitePlace K, f w = w : { w : InfinitePlace K // w.IsReal }, f w + w : { w : InfinitePlace K // w.IsComplex }, f w

                The map from real embeddings to real infinite places as an equiv

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                  The map from nonreal embeddings to complex infinite places

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                    @[simp]
                    theorem NumberField.InfinitePlace.mkReal_coe {K : Type u_2} [Field K] (φ : { φ : K →+* // ComplexEmbedding.IsReal φ }) :
                    (mkReal φ) = mk φ
                    @[simp]
                    theorem NumberField.InfinitePlace.prod_eq_abs_norm {K : Type u_2} [Field K] [NumberField K] (x : K) :
                    w : InfinitePlace K, w x ^ w.mult = |(Algebra.norm ) x|

                    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.

                    theorem NumberField.InfinitePlace.one_le_of_lt_one {K : Type u_2} [Field K] [NumberField K] {w : InfinitePlace K} {a : RingOfIntegers K} (ha : a 0) (h : ∀ ⦃z : InfinitePlace K⦄, z wz a < 1) :
                    1 w a
                    theorem NumberField.is_primitive_element_of_infinitePlace_lt {K : Type u_2} [Field K] [NumberField K] {x : RingOfIntegers K} {w : InfinitePlace K} (h₁ : x 0) (h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ww' x < 1) (h₃ : w.IsReal |(w.embedding x).re| < 1) :
                    x =
                    theorem NumberField.adjoin_eq_top_of_infinitePlace_lt {K : Type u_2} [Field K] [NumberField K] {x : RingOfIntegers K} {w : InfinitePlace K} (h₁ : x 0) (h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ww' x < 1) (h₃ : w.IsReal |(w.embedding x).re| < 1) :
                    @[reducible, inline]
                    noncomputable abbrev NumberField.InfinitePlace.nrRealPlaces (K : Type u_2) [Field K] [NumberField K] :

                    The number of infinite real places of the number field K.

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                      @[deprecated NumberField.InfinitePlace.nrRealPlaces (since := "2024-10-24")]

                      Alias of NumberField.InfinitePlace.nrRealPlaces.


                      The number of infinite real places of the number field K.

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                        @[reducible, inline]
                        noncomputable abbrev NumberField.InfinitePlace.nrComplexPlaces (K : Type u_2) [Field K] [NumberField K] :

                        The number of infinite complex places of the number field K.

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                          @[deprecated NumberField.InfinitePlace.nrComplexPlaces (since := "2024-10-24")]

                          Alias of NumberField.InfinitePlace.nrComplexPlaces.


                          The number of infinite complex places of the number field K.

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                            The infinite place of the rationals. #

                            The infinite place of , coming from the canonical map ℚ → ℂ.

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                              @[simp]