Theorems · Theorem · number theory
ClassGroup.exists_min
∀ {R : Type u_1} {S : Type u_2} [inst : EuclideanDomain R] [inst_1 : CommRing S] [IsDomain S] [inst_3 : Algebra R S]
(abv : AbsoluteValue R ℤ) (I : ↥(nonZeroDivisors (Ideal S))),
∃ b ∈ ↑I, b ≠ 0 ∧ ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0A nonzero ideal has an element of minimal norm.
- Defined in
- Mathlib.NumberTheory.ClassNumber.Finite
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 127 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
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
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Idealstatement and proof · cited by 4,748
- MonoidHomstatement · cited by 3,629
- Submonoidstatement · cited by 3,086
- IsDomainstatement and proof · cited by 2,196
- nonZeroDivisorsstatement and proof · cited by 895
- AbsoluteValuestatement and proof · cited by 363
- Algebra.normstatement and proof · cited by 155
- EuclideanDomainstatement and proof · cited by 124
- Submodule.ne_bot_iffproof · cited by 28
Cited by1
Results whose statement or proof uses this declaration.
- ClassGroup.exists_mk0_eq_mk0proof · cited by 1