Dedekind's criterion for the monogenicity of a number field versus Uchida's and L\"uneburg's

We compare three different characterizations, due respectively to R. Dedekind, K. Uchida, and H. L"uneburg, of when $\mathbb{Z}[\theta ]$ is the ring of integers of $\mathbb{Q}(\theta )$, and apply our results to some concrete $2$-towers of number fields.