Krull's original paper on infinite Galois extensions,
Math. Ann. 100, 687-698 (1928);
Baer's review of Krull's paper in Zentralblatt
I found this very interesting. For example, it seems "separable" was called "algebraic of first kind". Also keep
in mind that Hausdorff's definition of a topological space, which included the Hausdorff axiom, wasn't much more than 10 years old at that time.
It seems Krull was
initially mainly interested in the case of countable infinite extensions (and in that case the Galois group is metrisable); he thanks
John v. Neumann for the observation that this should generalise (see footnote 11).
Note that the "finite cover" notion of compactness wasn't introduced until 1929, which explains why Krull only shows that Galois groups are compact in the countable case.
Finally he discusses the case of
an "ideal-cyclic" (i.e., procyclic) Galois group.
Krull mentions that Dedekind (1901) was the first to realise that intermediate fields and subgroups of the Galois group aren't in natural
bijection for infinite extensions. Dedekind's example was the extension of the rationals obtained by adjoining pn-th roots of unity for all n,
for a fixed prime p. (The review of Dedekind's paper at Zentralblatt is almost incomprehensible in its old-fashioned mathematical language...)