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

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