  The problem of the "common inessential discriminant divisors" attracted the
attention of Dedekind, Kronecker, and Hensel in the early days of algebraic
number theory. Four sources are particularly important: Dedekind's
announcement, in 1871, of the second edition of Dirichlet's lectures
\cite{anzeige}, Dedekind's 1878 paper, the 25th section of Kronecker's 1882
\textit{Grundz\"uge}, and Hensel's 1894 paper, which is our focus here. Both of
the key papers of Dedekind were translated and annotated in our paper "Dedekind
on higher congruences and index divisors, 1871 and 1878." (arXiv:2107.08905}.
We here present an annotated translation of Kurt Hensel's "Arithmetische
Untersuchungen \"uber die gemeinsamen ausserwesentlichen Discriminantentheiler
einer Gattung" (\textit{Journal f\"ur die Reine und Angewandte Mathematik},
\textbf{113} (1894), 128--160).
