A number field is $\textit{monogenic}$ (over $\mathbb{Q}$) if the ring of integers admits a power integral basis, i.e., a $\mathbb{Z}$-basis of the form $\{1, \alpha, \alpha^2,\dots, \alpha^{n-1}\}$. In this case we call $\alpha$ a $\textit{monogenerator}$. The first portion of the talk will be spent revisiting some classical examples of monogenicity and non-monogenicity. We will touch on some recent work on radical extensions while paying particular attention to obstructions to monogenicity and relations to other arithmetic questions. The latter part of the talk will be devoted to recent work ([1] and [2]) constructing a general moduli space of monogenerators. Specifically, given an extension of algebras $B/A$, we construct a $\textit{scheme}$ $\mathcal{M}_{B/A}$ parameterizing the possible choices of a monogenerator for $B$ over $A$.
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$$\textbf{References} $$
[1] Arpin, S., Bozlee, S., Herr, L., and Smith, H. (2021). The Scheme of Monogenic Generators I: Representability.
arXiv: https://arxiv.org/abs/2108.07185. (Accepted to Research in Number Theory.)
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[2] Arpin, S., Bozlee, S., Herr, L., and Smith, H. (2022). The Scheme of Monogenic Generators II: Local Monogenicity and Twists.
arXiv: https://arxiv.org/abs/2205.04620.
