Theorems · Theorem · number theory
NumberField.exists_not_isUnramifiedAt_int
∀ {K : Type u_1} {𝒪 : Type u_2} [inst : Field K] [inst_1 : NumberField K] [inst_2 : CommRing 𝒪] [inst_3 : Algebra 𝒪 K]
[IsIntegralClosure 𝒪 ℤ K], Module.finrank ℚ K ≠ 1 → ∃ P, ∃ (x : P.IsMaximal), ¬Algebra.IsUnramifiedAt ℤ PIf K is a number field with positive rank, then there exists some maximal ideal of 𝓞 K
that is ramified over ℤ.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 318 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Idealstatement · cited by 4,748
- Algebra.algebraMapproof · cited by 4,706
- IsDomainproof · cited by 2,196
- Nat.Primeproof · cited by 2,059
- Module.finrankstatement and proof · cited by 1,770
- Ideal.spanproof · cited by 948
- IsDedekindDomainproof · cited by 668
- NumberFieldstatement and proof · cited by 653
- Module.IsTorsionFreeproof · cited by 600
Cited by2
Results whose statement or proof uses this declaration.
- NumberField.finrank_eq_one_of_unramifiedproof · cited by 1
- NumberField.exists_not_isUnramifiedAt_int_of_isGaloisproof · cited by 0