Addendum: Infinite Galois Correspondence

The consideration of algebraic extensions is fundamental in arithmetic. We will see that the Galois correspondence (Theorem 6.5.1 ) generalizes as long as we equip the Galois group with a suitable topology, called the profinite topology. In the following, \(K/k\) denotes a Galois extension of perfect fields, \(\mathcal {F}\) the family of subfields L of K containing k and \({\mathcal {F}_{in}}\) the subfamily of \(\mathcal {F}\) such that L is Galois of finite degree over k. They are ordered by inclusion. We assume the reader is familiar with the basic definitions of general topology.