Theorems · Theorem · field theory
minpoly.natSepDegree_eq_one_iff_pow_mem
∀ {F : Type u} {E : Type v} [inst : Field F] [inst_1 : Ring E] [IsDomain E] [inst_3 : Algebra F E] (q : ℕ)
[hF : ExpChar F q] {x : E}, (minpoly F x).natSepDegree = 1 ↔ ∃ n, x ^ q ^ n ∈ (algebraMap F E).rangeThe minimal polynomial of an element x of E / F of exponential characteristic q has
separable degree one if and only if x ^ (q ^ n) ∈ F for some n : ℕ.
- Defined in
- Mathlib.FieldTheory.SeparableDegree
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites30
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Algebra.algebraMapstatement and proof · cited by 4,706
- IsDomainstatement and proof · cited by 2,196
- Polynomial.Xproof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- Polynomial.aevalproof · cited by 615
- Subringstatement · cited by 602
- map_subproof · cited by 565
- LE.le.antisymmproof · cited by 507
Cited by4
Results whose statement or proof uses this declaration.
- isPurelyInseparable_iff_pow_memproof · cited by 10
- minpoly.natSepDegree_eq_one_iff_eq_X_sub_C_powproof · cited by 1
- isPurelyInseparable_iff_natSepDegree_eq_oneproof · cited by 1
- mem_perfectClosure_iff_natSepDegree_eq_oneproof · cited by 1