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In Brian Osserman's notes on infinite Galois theory, in the third paragraph of page 3 (proof of the fundamental theorem of Infinite Galois Theory) he says "let $E/F$" be THE Galois closure. I am not sure exactly what he means. As i understand, there exist many Galois closures of a field $F$. For example, if $K/F$ is a Galois extension, then for any $a \in K$ let $F_a$ be the splitting field of $a$ over $F$. Then $F_a/F$ is a finite Galois closure. Additionally, for the argument that he wants to make, he needs $E/F$ to be finite, but he is not mentioning anything like that. Any comments?

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You are right: The Galois closure is not unique! But it can be made unique in the following way (which is what is done in these notes). If $K/F$ is a Galois extension, and $L/F$ is a sub-extension, then the Galois closure of $L$ in $K$ is the smallest Galois extension of $L$ contained in $K$. By requiring that everything takes place in $K$, we obtain a canonical copy of the Galois closure of $L$, which justifies calling it "the" Galois closure of $L$.

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  • $\begingroup$ Could you please elaborate on what you mean by "canonical copy of the Galois closure"? Also is the Galois closure finite? $\endgroup$ – Manos Sep 11 '13 at 20:44
  • $\begingroup$ @Manos, sorry, I just saw your comment. I mean that $K$ contains a field which is Galois over $L$ and minimal with this property, and it is unique: it is "the" Galois closure of $L$ in $K$. The Galois closure of $L$ in $K$ could be non-finite over $L$, but obviously it is finite if $K$ is finite over $F$. $\endgroup$ – Bruno Joyal Sep 27 '13 at 20:36

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