Theorems · Theorem · number theory
NumberField.classNumber_eq_one_iff
∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K],
NumberField.classNumber K = 1 ↔ IsPrincipalIdealRing (NumberField.RingOfIntegers K)The class number of a number field is 1 iff the ring of integers is a PID.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldNumberField
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fieldstatement and proof · cited by 7,404
- NumberFieldstatement and proof · cited by 653
- NumberField.RingOfIntegersstatement · cited by 413
- IsPrincipalIdealRingstatement · cited by 131
- NumberField.classNumberstatement · cited by 7
- card_classGroup_eq_one_iffproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- RingOfIntegers.isPrincipalIdealRing_of_isPrincipal_of_norm_leproof · cited by 2
- Rat.classNumber_eqproof · cited by 0