Let F/k be an algebraic extension. Let S(F/k) be the set of all embeddings of F over k into algebraic closure ka. I'm trying to prove that the smallest normal extension of k containing F is
E=∏σ∈S(F/k)σF (Π is used to denote the compositum of fields)
In the finite extension case, matter is simple as if τ is embedding of E over k, σ↦τσ is an injective mapping of S(F/k) into S(F/k). Since S(F/k) is finite, the above mapping is bijective and E=∏σ∈S(F/k)σF=∏τσF.
But in the infinite extension case when S(F/k) might not be finite, surjective part must be added to prove that τ induces permutation on S(F/k). I don't quite get an idea of how I should prove the surjectiveness.
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