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Algebraic number theory is an ancient subject and its content is vast. Here, we make a very limited selection from the fascinating array of possible topics. Bridging the gap between the systematic study of some advanced topics concerning local fields in general case and practical themes on finite p-adic extensions. While some results presented are well or less known to the expert, some others novel results, typically original, are scattered throughout the text. Furthermore the whole work is simply but emphatically presented. Applications of the theory and even suggestions are included. The first part concerns local fields in the more general case that is without any assumption on the residue field. The two following parts are a study of some "standard over-finite-extensions", including some special results on the discriminant of such extensions. We too give a "Normality Criteria" based on the coefficients of Eisenstein polynomials, for the degrees p and p2. The last part summarizes some of the basic machinery in Algebra that is needed. We present a well developed and accessible text as instructive as imparting knowledge for the researcher as well as for advanced students.