Theorems · Theorem · number theory

NumberField.FinitePlace.norm_le_one

∀ (K : Type u_1) [inst : Field K] {R : Type u_2} [inst_1 : CommRing R] [inst_2 : Algebra R K]
  [inst_3 : IsDedekindDomain R] [inst_4 : IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
  [inst_5 : Module.Finite ℤ R] [inst_6 : Module.Free ℤ R] (x : R),
  ‖(NumberField.FinitePlace.embedding v) ((algebraMap R K) x)‖ ≤ 1

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

Defined in: Mathlib.NumberTheory.NumberField.Completion.FinitePlace
Cited by: 1 results in Mathlib
Foundations: Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field CommRing Algebra IsDedekindDomain IsFractionRing Module.Finite Module.Free

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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
Realstatement and proof · cited by 25,697
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
RingHomstatement · cited by 10,189
Fieldstatement and proof · cited by 7,404
Norm.normstatement · cited by 5,413
Algebra.algebraMapstatement and proof · cited by 4,706
Module.Finitestatement and proof · cited by 1,032
IsFractionRingstatement and proof · cited by 738
IsDedekindDomainstatement and proof · cited by 668
Module.Freestatement and proof · cited by 597

Cited by1

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NumberField.FinitePlace.hasFiniteMulSupport_intproof · cited by 3