Lemma 9.14.6. Let E/F be an algebraic field extension. There exists a unique subextension E/E_{sep}/F such that E_{sep}/F is separable and E/E_{sep} is purely inseparable.
Any algebraic field extension is uniquely a separable field extension followed by a purely inseparable one.
Proof. If the characteristic is zero we set E_{sep} = E. Assume the characteristic is p > 0. Let E_{sep} be the set of elements of E which are separable over F. This is a subextension by Lemma 9.12.13 and of course E_{sep} is separable over F. Given an \alpha in E there exists a p-power q such that \alpha ^ q is separable over F. Namely, q is that power of p such that the minimal polynomial of \alpha is of the form P(x^ q) with P separable algebraic, see Lemma 9.12.1. Hence E/E_{sep} is purely inseparable. Uniqueness is clear. \square
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