Lemma 9.15.4. Let E/F be a normal algebraic field extension. Then the subextension E/E_{sep}/F of Lemma 9.14.6 is normal.
Proof. If the characteristic is zero, then E_{sep} = E, and the result is clear. If the characteristic is p > 0, then E_{sep} is the set of elements of E which are separable over F. Then if \alpha \in E_{sep} has minimal polynomial P write P = c(x - \alpha )(x - \alpha _2) \ldots (x - \alpha _ d) with \alpha _2, \ldots , \alpha _ d \in E. Since P is a separable polynomial and since \alpha _ i is a root of P, we conclude \alpha _ i \in E_{sep} as desired. \square
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