Dedekind on Higher Congruences and Index Divisors, 1871 and 1878

Dedekind's theorem connecting ideal theory and polynomial congruences appears in all textbooks on algebraic number theory, but few books note its connection to the problem of ``common index divisors.'' As part of a project to study the history of this problem, we present an annotated translation of two of Dedekind's papers on the subject: a notice about the first publication of Dedekind's ideal theory in 1871 and a paper of 1878 giving proofs of the results announced in 1871 and giving a necessary and sufficient condition for the existence of common index divisors. A separate paper will analyze Hensel's 1894 paper containing the same theorem.