Valued fields: p-adic numbers as a case study

Abstract

In this thesis, we first build the idea of field valuation and look into some of its examples, followed by the idea of general absolute value, as they are closely related. Then, an absolute value induces a metric, and we can talk about completions of fields with respect to the induced metric. In particular, we examine in detail the field of p-adic numbers as a great example. Finally, we return to general valued fields, and extract this process entirely onto the valuation level, bypassing absolute value and induced metric. Most importantly, we want to apply Hensel’s lemma on general valued fields, and in the end we shall see when it is possible.