This book is accessible to graduate students, and can be used as a reference source by researcher mathematicians in algebra and number theory. It has four parts: Basic results on discrete valuation rings, Dedekind domains, and completions Ramification theory: discriminant, different, ramification subgroups, Hasse-Arf theorem, Artin representations Group cohomology, with emphasis on arithmetical applications: theorems of Tate and Nakayama, Galois cohomology, class informations Local class field theory, presented from the cohomological point of view. The main result is the determination of the topological Galois group of the maximal abelian extension