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146 7. Galois Cohomology and Isomorphism Classification of Elliptic Curves

F = lim K.
−→
K/F finite normal
The union of all finite separable subextensions is a subfield F s of F which is a Galois
extension of F contained in F. This is also the direct limit

F s = lim K.
−→
K/F finite Galois
The Galois group Gal(F s /F) maps to each Gal(K/F), where K is a finite Galois
subextension of F s , and we have an isomorphism onto the projective limit of finite
groups

Gal(F s /F) → lim Gal(K/F).
←−
K/F finite Galois
This projective limit has the limit topology in which it is compact and this compact
topology is transferred to Gal(F s /F) making the Galois group a compact topological
group.

(1.8) Remark. The Galois correspondence of (1.3) becomes the assertion that the
function which assigns to a closed subgroup H of Gal(F s /F) the subfield Fix(H) of
F s containing F is a bijection from the set of closed subgroups of Gal(F s /F) onto
the set of subfields of F s containing F.
(1.9) Example. For F = F q the finite field of q elements there exists exactly one
extension (up to isomorphism) of degree n over F q , namely F q . It is a Galois ex-
n
tension with cyclic Galois group Z/n with 1 in Z/n corresponding to the Frobenius
q
ˆ
automorphism a → q . The Galois group Gal(F q /F q ) is Z topologically gener-
ˆ
ated by the Frobenius automorphism, and Z is lim n Z/n, the completion of Z for the
←−
topology given by subgroups of finite index.
§2. Group Actions on Sets and Groups

We consider three types of objects: (1) sets, (2) pointed sets, i.e., sets with base point
∗, or (3) groups. For such an object E we have the group Aut(E) of automorphism
which:
(1) for a set E consists of all bijections E → E, i.e., permutations of E;
(2) for a pointed set E consists of all bijections preserving the base point; and
(3) for a group E consists of all group automorphisms.

(2.1) Definition. Let G be a group, and let E be an object, i.e., a set, a pointed set,
or a group. A left group action of G on E is a homomorphism G → Aut(E).