Theorems · Theorem · field theory
IsAlgebraic.exists_nonzero_coeff_and_aeval_eq_zero
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : Ring S] [inst_2 : Algebra R S] {s : S},
IsAlgebraic R s → s ∈ nonZeroDivisors S → ∃ q, q.coeff 0 ≠ 0 ∧ (Polynomial.aeval s) q = 0- Defined in
- Mathlib.RingTheory.Algebraic.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
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
- Ringstatement and proof · cited by 7,463
- Polynomialstatement and proof · cited by 5,681
- AlgHomstatement · cited by 3,236
- Submonoidstatement · cited by 3,086
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- map_mulproof · cited by 1,137
- Polynomial.coeffstatement · cited by 1,045
- sub_zeroproof · cited by 938
Cited by2
Results whose statement or proof uses this declaration.
- IsAlgebraic.exists_nonzero_dvdproof · cited by 5
- IsAlgebraic.exists_nonzero_eq_adjoin_mulproof · cited by 1