Topics in higher ramification theory

We introduce and study several notions in the setting of higher ramification theory, in particular ramification ideals and differents. After general results on the computation of ramification ideals, we discuss their connection with defect and compute them for Artin-Schreier extensions and Kummer extensions of prime degree equal to the residue degree, with or without defect. We present an example that shows that nontrivial defect in an extension of degree not a pime may not imply the existence of a nonprincipal ramification ideal. We compute differents for the mentioned extensions of prime degree, after computing the necessary traces, and discuss the question when they are equal to the annihilator of the Kähler differentials of the extension. Further, we introduce and study the ideal generated by the differents of the elements of the upper valuation rings in such extensions.