Degree p Extensions of Arbitrary Valuation Rings and "Best f"

We prove the explicit characterization of the so-called "best f" for degree p Artin-Schreier and degree p Kummer extensions of Henselian valuation rings in residue characteristic p. This characterization is mentioned briefly in [Th16, Th18]. Existence of best f is closely related to the defect of such extensions and this characterization plays a crucial role in understanding their intricate structure. We also treat degree p Artin-Schreier defect extensions of higher rank valuation rings, extending the results in [Th16], and thus completing the study of degree p extensions that are the building blocks of the general theory.