 Both Dedekind and Kronecker knew this earlier work well, especially Kummer’s, and they tried to build on it. Kronecker’s strategy was to examine in concrete detail, and by exploiting computational aspects, certain kinds of extensions. Crucial for him was to proceed finitistically and constructively, thus in a self-consciously restricted way. This led to his “divisor theory”, an extension of Gauss’ and Kummer’s “theory of forms” (Edwards 1980, 1990). Dedekind, in contrast, approached the issue in a more encompassing and abstract way. He considered algebraic number fields in general, thereby introducing the notion of a “field” for the first time (Edwards 1983, Haubrich 1992, Stillwell 1996, 1999, Ullrich 2017, Gray 2018).