We present a translation of a portion of Dedekind's Supplement XI to
Dirichlet's Vorlesungen über Zahlentheorie which contains an
investigation of the subfields of the complex numbers. 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 the complex numbers), in
stark contrast to the picture presented by Kiernan.
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.