Documentation

Mathlib.NumberTheory.NumberField.FinitePlaces

Finite places of number fields #

This file defines finite places of a number field K as absolute values coming from an embedding into a completion of K associated to a non-zero prime ideal of š“ž K.

Main Definitions and Results #

Tags #

number field, places, finite places

The norm of a maximal ideal is > 1

The norm of a maximal ideal as an element of ā„ā‰„0 is > 1

The norm of a maximal ideal as an element of ā„ā‰„0 is ā‰  0

The v-adic absolute value on K defined as the norm of v raised to negative v-adic valuation.

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    The embedding of a number field inside its completion with respect to v.

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      A finite place of a number field K is a place associated to an embedding into a completion with respect to a maximal ideal.

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        Return the finite place defined by a maximal ideal v.

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          The norm of the image after the embedding associated to v is equal to the v-adic absolute value.

          The norm of the image after the embedding associated to v is equal to the norm of v raised to the power of the v-adic valuation.

          The norm of the image after the embedding associated to v is equal to the norm of v raised to the power of the v-adic valuation for integers.

          The v-adic norm of an integer is at most 1.

          The v-adic norm of an integer is 1 if and only if it is not in the ideal.

          The v-adic norm of an integer is less than 1 if and only if it is in the ideal.

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          For a finite place w, return a maximal ideal v such that w = finite_place v .

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            @[simp]
            theorem NumberField.FinitePlace.mk_eq_iff {K : Type u_1} [Field K] [NumberField K] {vā‚ vā‚‚ : IsDedekindDomain.HeightOneSpectrum (RingOfIntegers K)} :
            mk vā‚ = mk vā‚‚ ā†” vā‚ = vā‚‚

            The equivalence between finite places and maximal ideals.

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              theorem NumberField.FinitePlace.mulSupport_finite_int {K : Type u_1} [Field K] [NumberField K] {x : RingOfIntegers K} (h_x_nezero : x ā‰  0) :
              (Function.mulSupport fun (w : FinitePlace K) => w ā†‘x).Finite
              theorem NumberField.FinitePlace.mulSupport_finite {K : Type u_1} [Field K] [NumberField K] {x : K} (h_x_nezero : x ā‰  0) :