Norm-one algebraic torus ($R_{L/k}^{(1)}{\mathbb{G}}_m$)

For a finite separable extension $L/k$ of degree $d$, the norm-one algebraic torus isHere $R_{L/k}$ is Weil restriction and the map extends the field norm. Its $k$-points are $\{x\in L^{\times }:N_{L/k}(x)=1\}$. The finite separability hypothesis makes it an algebraic torus of dimension $d-1$.

Indeed, after extending to a separable closure, $L$ splits into $d$ copies of the base field and the norm becomes $(z_1,\dots ,z_d)\mapsto z_1\cdots z_d$. Its kernel is isomorphic to ${\mathbb{G}}_m^{d-1}$ by choosing the first $d-1$ coordinates and taking the last as their inverse product.

For $L=\mathbb{C},k=\mathbb{R}$, the norm is $x^2+y^2$, giving the real unit circle with complex multiplication. It becomes a split one-dimensional torus over $\mathbb{C}$, although it is nonsplit over $\mathbb{R}$. Surjectivity of the norm as an algebraic-group morphism does not assert surjectivity on real points: complex norms are positive real numbers.