Theorems · Theorem · field theory
IntermediateField.isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one
∀ (F : Type u) (E : Type v) [inst : Field F] [inst_1 : Field E] [inst_2 : Algebra F E] {x : E},
IsPurelyInseparable F ↥F⟮x⟯ ↔ (minpoly F x).natSepDegree = 1F⟮x⟯ / F is a purely inseparable extension if and only if the minimal polynomial of x
has separable degree one.
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- IntermediateFieldstatement · cited by 988
- minpolystatement and proof · cited by 439
- IntermediateField.adjoinstatement and proof · cited by 382
- IsPurelyInseparablestatement · cited by 84
- Polynomial.natSepDegreestatement and proof · cited by 53
- IntermediateField.adjoin_simple_le_iffproof · cited by 9
- le_perfectClosure_iffproof · cited by 2
- mem_perfectClosure_iff_natSepDegree_eq_oneproof · cited by 1
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