An overview over the Infinite Galois Theory and Absolute Galois Groups

In this thesis the generalization of classical Galois Theory to the case of infinite algebraic extensions is analyzed. The key point is the notion of Krull topology on groups, providing a Fundamental Theorem of Galois Theory for infinite extensions and suitable topological groups. Inverse limits and profinite groups will be introduced and studied in an attempt to obtain a correspondence between such groups and arbitrary Galois extensions. Furthermore, two examples of infinite extensions will be provided for the purpose of studying their associated Galois groups. These examples pertain to the Absolute Galois group of the field of p-adic numbers and to finite fields of prime order, prior to which a brief introduction of local rings and ramification theory is provided