We establish the Artin-Schreier-Witt theory in connection with the unit group scheme of a group algebra, following a method presented by Serre in Groupes algébriques et corps de classes. The argument is developed not only over a field but also over a ring, as generally as possible.
"Artin-Schreier-Witt extensions and normal bases." Hiroshima Math. J. 42 (3) 325 - 354, November 2012. https://doi.org/10.32917/hmj/1355238372