Norm map for a commutative torsor with a separable point

By [F1], write $L=k(\alpha )$ and take a splitting field $K\mathrm{/}k$ of its separable minimal polynomial $m$, of degree $n$. This field is generated by finitely many algebraic roots, hence is finite and Galois by [F1]. The $n$ distinct roots of $m$ in $K$ correspond exactly to the $k$-embeddings $L=k[T]\mathrm{/}(m)\to K$, by evaluation at a root. Composing the point $\mathrm{Spec}L\to V$ with these embeddings gives $P_1,\dots ,P_n\in V(K)$, permuted by $\mathrm{Gal}(K\mathrm{/}k)$. The torsor isomorphism, restricted to the fibre whose first coordinate is $P_i$, identifies $G_K$ with $V_K$ by $g\mapsto P_i+g$. Its inverse is a morphism ${\varphi }_i:V_K\to G_K$, characterized by $P_i+{\varphi }_i(v)=v$. Thus ${\varphi }_i(v+g)={\varphi }_i(v)+g$ as a scheme morphism identity.

Put ${\varphi }_K={\sum }_{i=1}^n{\varphi }_i$, using the commutative group law. Semilinear Galois action permutes the summands and therefore preserves this morphism. By [F2], ${\varphi }_K$ descends to $\varphi :V\to G$. Summing the identities from step 1.1 gives ${\varphi }_K(v+g)={\varphi }_K(v)+ng$. Equality of these morphisms descends by the uniqueness in [F2], proving the required identity over $k$. The integer $n$ is positive; no division by $n$ or characteristic restriction is made. If $V$ is nonempty and smooth, [F3] supplies the required point $P$, completing that clause as well. AC is inherited from [F3] when that clause is used.