Theorems · Theorem · field theory
Polynomial.Separable.squarefree
∀ {R : Type u} [inst : CommSemiring R] {p : Polynomial R}, p.Separable → Squarefree pA separable polynomial is square-free.
See PerfectField.separable_iff_squarefree for the converse when the coefficients are a perfect
field.
- Defined in
- Mathlib.FieldTheory.Separable
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Polynomialstatement and proof · cited by 5,681
- IsUnitproof · cited by 1,602
- Polynomial.Separablestatement and proof · cited by 117
- Squarefreestatement · cited by 112
- squarefree_iff_emultiplicity_le_oneproof · cited by 4
- Polynomial.emultiplicity_le_one_of_separableproof · cited by 3
Cited by6
Results whose statement or proof uses this declaration.
- Module.End.IsSemisimple.of_mem_adjoin_pairproof · cited by 3
- PerfectField.separable_iff_squarefreeproof · cited by 2
- Polynomial.squarefree_cyclotomicproof · cited by 1
- Module.End.exists_isNilpotent_isSemisimple_of_separable_of_dvd_powproof · cited by 1
- IsPrimitiveRoot.minpoly_eq_powproof · cited by 1
- IsPrimitiveRoot.squarefree_minpoly_modproof · cited by 1