This thesis is essentially a compendium of the author's completed work as a graduate student. It includes a variety of different investigations centering around monogeneity and division fields. In the defense, we will introduce these concepts with some historical backing. The speaker hopes this portion will be friendly to a relatively general audience. After moving through some classical results, we will discuss the author's earlier work (joint with Katherine Stange and T. Alden Gassert) on the monogeneity of partial torsion fields. This segues into the author's work on the monogeneity of quartic fields generated by trinomials. We will move to more recent work on the radical extensions. Finally, we will discuss results on the non-monogeneity of division fields of certain abelian varieties, including explicit non-monogenic families in the case of dimension 1.
Everyone is invited. After the public presentation, the general audience will be asked to leave during private questioning, but a re-invite for a celebration/congratulations will be emailed out afterward, so please watch for that.