438 9. FIELD EXTENSIONS – SECOND LOOK The following is an important technical result that is used in the sequel. Proposition 9.5.6. Let K ± L be a ﬁnite–dimensional ﬁeld extension and let A;B be ﬁelds intermediate between K and L . Suppose that B is Galois over K . Then A ² B is Galois over A and Aut A .A ² B/ Š Aut A \ B .B/ . Proof. Exercise 9.5.2 . n The remainder of this section can be omitted without loss of continuity. It is not hard to obtain the inequality of Corollary 9.4.18 without the separability assumption. It follows that the separability assumption can also be removed in Proposition 9.5.3 . Although we consider only separable ﬁeld extensions in this text, the argument is nevertheless worth knowing. Proposition 9.5.7. Let L be a ﬁeld. Any collection of distinct automor-phisms of L is linearly independent. Proof. Let f ± 1 ;:::;± n g be a collection of distinct automorphisms of L . We show by induction on n that the collection is linearly independent (in
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