Mathlib Map

Theorems · Theorem · commutative algebra

card_classGroup_eq_one_iff

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] [IsDedekindDomain R] [inst_3 : Fintype (ClassGroup R)],
  Fintype.card (ClassGroup R) = 1 ↔ IsPrincipalIdealRing R

The class number is 1 iff the ring of integers is a principal ideal domain.

Defined in
Mathlib.RingTheory.ClassGroup.Basic
Cited by
2 results in Mathlib
Foundations
Depth 148 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
CommRingIsDomainIsDedekindDomainFintype
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

NumberField.classNumber_eq_one_iff · cited by 2NumberField.classNumber_e…FunctionField.classNumber_eq_one_iff · cited by 0FunctionField.classNumber…DFunLike.coe · cited by 62936DFunLike.coeCommRing · cited by 17173CommRingFintype · cited by 7736FintypeIdeal · cited by 4748IdealBot.bot · cited by 4720Bot.botIsDomain · cited by 2196IsDomainFintype.card · cited by 1386Fintype.cardIsDedekindDomain · cited by 668IsDedekindDomainIsPrincipalIdealRing · cited by 131IsPrincipalIdealRingSubmodule.IsPrincipal · cited by 129Submodule.IsPrincipalFintype.card_congr' · cited by 60Fintype.card_congr'ClassGroup · cited by 50ClassGroupmem_nonZeroDivisors_iff_ne_zero · cited by 38mem_nonZeroDivisors_iff_n…ClassGroup.mk0 · cited by 22ClassGroup.mk0Fintype.card_eq_one_iff · cited by 11Fintype.card_eq_one_iffcard_classGroup_eq_one_iffCITED BYCITES

Cites17

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by2

Results whose statement or proof uses this declaration.