English
The same construction yields a valid basis for any i-th component under exponent-char maps.
Русский
Та же конструкция образует валидный базис для каждой компоненты при отображении степеней.
LaTeX
$$$\text{basis}_i^{q^n}$ forms a basis$$
Lean4
/-- If `K / E / F` is a field extension tower, such that `E / F` is purely inseparable and `K / E`
is separable, then the separable degree of `K / F` is equal to the degree of `K / E`.
It is a special case of `Field.lift_sepDegree_mul_lift_sepDegree_of_isAlgebraic`, and is an
intermediate result used to prove it. -/
theorem sepDegree_eq_of_isPurelyInseparable_of_isSeparable [IsPurelyInseparable F E] [Algebra.IsSeparable E K] :
sepDegree F K = Module.rank E K :=
by
have h :=
(separableClosure F K).linearDisjoint_of_isPurelyInseparable_of_isSeparable
E |>.adjoin_rank_eq_rank_left_of_isAlgebraic_left |>.symm
rwa [separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable K, rank_top'] at h
