Theorems · Theorem · field theory
Polynomial.root_right_of_root_gcd
∀ {R : Type u} {k : Type y} [inst : Field R] [inst_1 : CommSemiring k] [inst_2 : DecidableEq R] {ϕ : R →+* k}
{f g : Polynomial R} {α : k}, Polynomial.eval₂ ϕ α (EuclideanDomain.gcd f g) = 0 → Polynomial.eval₂ ϕ α g = 0- Defined in
- Mathlib.Algebra.Polynomial.FieldDivision
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldCommSemiringDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommSemiringstatement and proof · cited by 10,911
- RingHomstatement and proof · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- MulZeroClass.zero_mulproof · cited by 1,625
- Polynomial.eval₂statement and proof · cited by 267
- EuclideanDomain.gcdstatement and proof · cited by 39
- Polynomial.eval₂_mulproof · cited by 30
- EuclideanDomain.gcd_dvd_rightproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- Field.primitive_element_inf_auxproof · cited by 1
- Polynomial.root_gcd_iff_root_left_rightproof · cited by 1