Theorems · Theorem · field theory
IsConjRoot.eq_zero_of_injective
∀ {R : Type u_1} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [IsDomain S]
[IsDomain R] [Module.IsTorsionFree R S] {x : S}, IsConjRoot R 0 x → Function.Injective ⇑(algebraMap R S) → x = 0A variant of IsConjRoot.eq_zero, only assuming IsDomain R,
IsTorsionFree R A and Function.Injective (algebraMap R A) instead of Field R. If x is a
conjugate root of 0, then x = 0.
- Defined in
- Mathlib.FieldTheory.Minpoly.IsConjRoot
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 136 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- RingHomstatement · cited by 10,189
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement and proof · cited by 600
- RingHom.map_zeroproof · cited by 47
- IsConjRootstatement and proof · cited by 43
- IsConjRoot.eq_algebraMap_of_injectiveproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- isConjRoot_zero_iff_eq_zero_of_injectiveproof · cited by 2
- IsConjRoot.ne_zero_of_injectiveproof · cited by 1
- IsConjRoot.eq_zeroproof · cited by 0