Degree One Milnor $K$-invariants of Groups of Multiplicative Type

Abstract: Let $G$ be a commutative affine algebraic group over a field $F$, and let $H \colon \mathrm{Fields}{F} \to \mathrm{AbGrps}$ be a functor. A (homomorphic) $H$-invariant of $G$ is a natural transformation $\mathrm{Tors}(-, G) \to H$, where $\mathrm{Tors}(-, G)$ is the functor $\mathrm{Fields}{F} \to \mathrm{AbGrps}$ taking a field extension $L/F$ to the group of isomorphism classes of $G_{L}$-torsors over $\mathrm{Spec}(L)$. The goal of this paper is to compute the group $\mathrm{Inv}{\mathrm{hom}}^{1}(G, H)$ of $H$-invariants of $G$ when $G$ is a group of multiplicative type, and $H$ is the functor taking a field extension $L/F$ to $L^{\times} \otimes{\mathbb{Z}} \mathbb{Q}/\mathbb{Z}$.