# Relationship Between Field Automorphisms and Embeddings

I'm reading some Galois theory in Lang's Algebra, and he often refers to maps acting on elements of a field extension as embeddings in the algebraic closure of the base field (if I'm not mistaken). However, it seems that the things he refers to as embeddings are replaced by elements of the Galois group in many other texts, for instance in the definition of norm and trace. Can someone tell me how these two things are related? I guess I can see it intuitively, but why when the extension is Galois do embeddings into the algebraic closure become automorphisms? Any help would be much appreciated.

Thanks!

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Let $q$ be in the extension. Let $f$ be the monic irreducible polynomial it satisfies over the base field. Let $T$ be the embedding. $T$ is a field homomorphism, and is the identity on the base field, so $0=T(0)=T(f(q))=f(T(q))$. Thus, $T(q)$ is a conjugate of $q$. –  Gerry Myerson Apr 20 '11 at 3:31