Theorems · Theorem · field theory

Polynomial.root_gcd_iff_root_left_right

∀ {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 f = 0 ∧ Polynomial.eval₂ ϕ α g = 0

Defined in: Mathlib.Algebra.Polynomial.FieldDivision
Cited by: 1 results in Mathlib
Foundations: Depth 123 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field CommSemiring DecidableEq

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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
Polynomial.eval₂statement and proof · cited by 267
EuclideanDomain.gcdstatement and proof · cited by 39
Polynomial.eval₂_gcd_eq_zeroproof · cited by 2
Polynomial.root_left_of_root_gcdproof · cited by 2
Polynomial.root_right_of_root_gcdproof · cited by 2

Cited by1

Results whose statement or proof uses this declaration.

Polynomial.isRoot_gcd_iff_isRoot_left_rightproof · cited by 0