The field extension Ks/K is infinite, and the Galois group is accordingly given the Krull topology.
is the profinite completion of integers with respect to its subgroups of finite index.
Another example is K = C((T)), the ring of formal Laurent series in T over the field C of complex numbers.
(These are simply formal power series in which we also allow finitely many terms of negative degree.)
Then K has a unique cyclic extension of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and that a generator of Gal(Kn/K) is given by This construction works if C is replaced by any algebraically closed field C of characteristic zero.