Theorems · Theorem · field theory
Polynomial.degree_eq_one_of_irreducible_of_root
∀ {R : Type u} [inst : CommRing R] {p : Polynomial R} [IsDomain R], Irreducible p → ∀ {x : R}, p.IsRoot x → p.degree = 1- Defined in
- Mathlib.Algebra.Polynomial.Div
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Polynomialstatement and proof · cited by 5,681
- add_zeroproof · cited by 2,707
- IsDomainstatement and proof · cited by 2,196
- Polynomial.Xproof · cited by 1,639
- IsUnitproof · cited by 1,602
- Polynomial.Cproof · cited by 1,598
- WithBotstatement and proof · cited by 1,498
- Polynomial.degreestatement and proof · cited by 643
- Irreduciblestatement and proof · cited by 496
- Polynomial.IsRootstatement and proof · cited by 152
Cited by7
Results whose statement or proof uses this declaration.
- IsAlgClosed.of_exists_rootproof · cited by 5
- Irreducible.not_isRoot_of_natDegree_ne_oneproof · cited by 2
- IsSepClosed.of_exists_rootproof · cited by 1
- Polynomial.roots_eq_zero_of_irreducible_of_natDegree_ne_oneproof · cited by 0
- Irreducible.subsingleton_isRootproof · cited by 0
- Polynomial.card_roots_le_one_of_irreducibleproof · cited by 0
- Polynomial.isIntegrallyClosed_iff'proof · cited by 0