Theorems · Theorem · commutative algebra
HenselianLocalRing.is_henselian
∀ {R : Type u_1} {inst : CommRing R} [self : HenselianLocalRing R] (f : Polynomial R),
f.Monic →
∀ (a₀ : R),
Polynomial.eval a₀ f ∈ IsLocalRing.maximalIdeal R →
IsUnit (Polynomial.eval a₀ (Polynomial.derivative f)) → ∃ a, f.IsRoot a ∧ a - a₀ ∈ IsLocalRing.maximalIdeal R- Defined in
- Mathlib.RingTheory.Henselian
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- HenselianLocalRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHom.idstatement · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- LinearMapstatement · cited by 10,215
- Polynomialstatement · cited by 5,681
- Idealstatement · cited by 4,748
- IsUnitstatement · cited by 1,602
- Polynomial.evalstatement · cited by 796
- Polynomial.Monicstatement · cited by 461
- Polynomial.derivativestatement · cited by 331
- IsLocalRing.maximalIdealstatement · cited by 297
- Polynomial.IsRootstatement · cited by 152
Cited by1
Results whose statement or proof uses this declaration.
- HenselianLocalRing.TFAEproof · cited by 0