Theorems · Theorem · field theory
IsAlgClosed.degree_eq_one_of_irreducible
∀ (k : Type u) [inst : Field k] [IsAlgClosed k] {p : Polynomial k}, Irreducible p → p.degree = 1If k is algebraically closed, then every irreducible polynomial over k is linear.
- Defined in
- Mathlib.FieldTheory.IsAlgClosed.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FieldIsAlgClosed
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement and proof · cited by 5,681
- WithBotstatement · cited by 1,498
- Polynomial.degreestatement · cited by 643
- Irreduciblestatement and proof · cited by 496
- IsAlgClosedstatement and proof · cited by 150
- IsAlgClosed.splitsproof · cited by 29
- Polynomial.Splits.degree_eq_one_of_irreducibleproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- IsAlgClosed.algebraMap_bijective_of_isIntegralproof · cited by 6
- Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_one_mul_isMonicOfDegreeproof · cited by 0