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Theorems · Theorem · number theory

NumberField.Units.rootsOfUnity_eq_torsion

∀ (K : Type u_1) [inst : Field K] [NumberField K],
  rootsOfUnity (NumberField.Units.torsionOrder K) (NumberField.RingOfIntegers K) = NumberField.Units.torsion K

The group of roots of unity of order dividing torsionOrder is equal to the torsion group.

Defined in
Mathlib.NumberTheory.NumberField.Units.Basic
Cited by
2 results in Mathlib
Foundations
Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
FieldNumberField
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Ideal.torsionMapQuot_injective · cited by 1Ideal.torsionMapQuot_inje…NumberField.Units.map_complexEmbedding_torsion · cited by 1Units.map_complexEmbeddin…Field · cited by 7404FieldSubgroup · cited by 3593SubgroupUnits · cited by 2804UnitsNumberField · cited by 653NumberFieldNumberField.RingOfIntegers · cited by 413NumberField.RingOfIntegersrootsOfUnity · cited by 118rootsOfUnitySubgroup.ext · cited by 108Subgroup.extNumberField.Units.torsion · cited by 53Units.torsionisOfFinOrder_iff_pow_eq_one · cited by 27isOfFinOrder_iff_pow_eq_o…NumberField.Units.torsionOrder · cited by 17Units.torsionOrderCommGroup.torsion · cited by 16CommGroup.torsionmem_rootsOfUnity · cited by 10mem_rootsOfUnitypow_card_eq_one' · cited by 9pow_card_eq_one'CommGroup.mem_torsion · cited by 6CommGroup.mem_torsionNumberField.Units.torsionOrder_pos · cited by 4Units.torsionOrder_posUnits.rootsOfUnity_eq_torsionCITED BYCITES

Cites15

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Cited by2

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