  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.
