A number field $K$ is monogenic if the ring of integers admits a power integral basis, i.e., if ${\mathcal{O}}_{K}=\mathbb{Z}[\alpha ]$ for some $\alpha \in {\mathcal{O}}_{K}$. This talk will survey recent results on monogeneity with a focus on the speaker's recent results on radical extensions and torsion fields of abelian varieties.

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Motivic homotopy theory over R is interesting in part because of its connection to ordinary stable homotopy theory and to C_2-equivariant homotopy theory. In this talk I will review some of these connections, and discuss work in progress with Dan Isaksen to compute R-motivic stable homotopy groups of spheres using an Adams spectral sequence. One of our main applications is to a variant of the Mahowald invariant which can be computed using knowledge of the R-motivic Adams spectral sequence.