Theorems · Theorem · number theory

RingOfIntegers.isPrincipalIdealRing_of_isPrincipal_of_norm_le_of_isPrime

∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K],
  (∀ ⦃I : ↥(nonZeroDivisors (Ideal (NumberField.RingOfIntegers K)))⦄,
      (↑I).IsPrime →
        ↑(Ideal.absNorm ↑I) ≤
            (4 / Real.pi) ^ NumberField.InfinitePlace.nrComplexPlaces K *
              (↑(Module.finrank ℚ K).factorial / ↑(Module.finrank ℚ K) ^ Module.finrank ℚ K *
                √|↑(NumberField.discr K)|) →
          Submodule.IsPrincipal ↑I) →
    IsPrincipalIdealRing (NumberField.RingOfIntegers K)

Defined in: Mathlib.NumberTheory.NumberField.ClassNumber
Cited by: 1 results in Mathlib
Foundations: Depth 315 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field NumberField

Mathlib docs Source Loogle LeanSearch

Around this declaration

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

Cites36

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 · cited by 25,697
Fieldstatement and proof · cited by 7,404
Idealstatement and proof · cited by 4,748
LE.le.transproof · cited by 3,151
Submonoidstatement · cited by 3,086
absstatement and proof · cited by 1,814
Real.pistatement and proof · cited by 1,774
Module.finrankstatement and proof · cited by 1,770
nonZeroDivisorsstatement and proof · cited by 895
Ideal.IsPrimestatement and proof · cited by 827
MonoidWithZeroHomstatement · cited by 704

Cited by1

Results whose statement or proof uses this declaration.

RingOfIntegers.isPrincipalIdealRing_of_isPrincipal_of_pow_le_of_mem_primesOver_of_mem_Iccproof · cited by 1