Dedekind's Treatment of Galois Theory in the Vorlesungen

We present a translation of §§160–166 of Dedekind’s Supplement XI to Dirichlet’s Vorlesungen über Zahlentheorie, which contain an investigation of the subfields of C. In particular, Dedekind explores the lattice structure of these subfields, by studying isomorphisms between them. He also indicates how his ideas apply to Galois theory. After a brief introduction, we summarize the translated excerpt, emphasizing its Galois-theoretic highlights. We then take issue with Kiernan’s characterization of Dedekind’s work in his extensive survey article on the history of Galois theory; Dedekind has a nearly complete realization of the modern “fundamental theorem of Galois theory” (for subfields of C), in stark contrast to the picture presented by Kiernan at points. We intend a sequel to this article of an historical and philosophical nature. With that in mind, we have sought to make Dedekind’s text accessible to as wide an audience as possible. Thus we include a fair amount of background and exposition.