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?
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 $L$.