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We recall that a number field $K$ is called monogenic if there exists some $\alpha \in \mathcal{O}_K$ such that $\mathcal{O}_K = \mathbb{Z}[\alpha]$.

If instead we take a tower of finite extensions $L-K-\mathbb{Q}$, it seems reasonable to call the extension $L/K$ monogenic if there exists some $\alpha \in \mathcal{O}_L$ such that $\mathcal{O}_L = \mathcal{O}_K[\alpha]$. In particular, this recovers our original definition when $K=\mathbb{Q}$. Monogenic field extensions also have the property that the relative discriminant $\Delta_{L/K}$ is principal, generated by the discriminant of the minimal polynomial of $\alpha$ over $K[t]$.

I've looked for references to this generalization without success, but would be grateful for one. In particular, I'm looking for an example of a monogenic field extension $L/K$ such that $\Delta_{L/K} = \mathcal{O}_K$ (this last condition is equivalent to saying that no prime ideal in $K$ ramifies over $L$). Note: Minkowski's Theorem implies that this is impossible for $K = \mathbb{Q}$.

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This is in Chapter 3 of Serre's Corps Locaux (Local fields) and also in Chapter 3.2 of Neukirch's Algebraic Number theory.