Asumptions required for Inverse Limit on Infinite Galois Group I am currently trying to get some knowledge on Infinite Galois Theory.
During the construction of the Galois Group of an infinite Galois extension through an inverse limit of Galois group of intermediary finite Galois extensions, there is one point that puzzles me.
Some authors require the inverse system of the inverse limit to be based on a directed set (for example Milne), while some authors only require the weaker asumption of a partially ordered set.
So for example in Milne's approach we have to check that the compositum EF of two finite galois extensions E and F is still a finite galois extension before we are able to talk about the inverse limit.
Do we gain anything from this extra property of the inverse limit in this context, or is it just a difference of convention used by different authors ?
 A: For an infinite normal extensions $E/F$, for each $a\in E$ take a finite normal extension $K_a/F$ such that $a\in K_a$. For simplicity take $K_a$ the normal closure of $F(a)/F$.
Then $Aut(E/F)$ is the group $G$ whose elements are maps $f:E\to \bigcup_a Aut(K_a/F)$ such that $f(a)\in Aut(K_a/F),f(a)|_{K_a\cap K_b}=f(b)|_{K_a\cap K_b}$.
It is immediate that $Aut(E/F)$ injects naturally into $G$.
It remains to check that every element of $G$ defines an automorphism on the whole of $E$. For this you need to show that $f(a),f(b)$ gives an automorphism of $K_aK_b$.
As you see there is nothing abstract here.
If $E/F$ is separable then the primitive element theorem gives $K_aK_b\subset K_c$ and $f(c)|_{K_aK_b}$  is our automorphism. When $E/F$ is not separable you can replace $E$ by $E\cap F^{sep}$ as any automorphism of $E\cap F^{sep}/F$ extends easily to $E/F$.
A: Doesn't matter. Define a profinite group to be a projective limit of finite groups, each with the discrete topology. Whether or not you require the projective limit to be over a directed set, you find that a profinite group is compact and totally disconnected. But a compact totally disconnected group is a projective limit of finite groups over a directed set, namely, over the set of its open normal subgroups, which is directed because the intersection of two open normal subgroups is open normal. See Serre, Galois Cohomology, first page.
