Why does the group of automorphisms of a field extension K/F sometimes fail to coincide with the Galois group of K over F?

💡 Explanation

The automorphism group might not equal the Galois group because normality ensures that every embedding of K into an algebraic closure of F maps K into itself, therefore defining an automorphism. If K/F is not normal, embeddings into the algebraic closure do not necessarily restrict to automorphisms of K, rather they map into a larger field.

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