= Norm-one algebraic torus
{title2=$R^{(1)}_{L/k}\mathbb G_m$}
For a finite <separable extension> $L/k$ of degree $d$, the norm-one algebraic torus is
$$
R^{(1)}_{L/k}\mathbb G_m=\ker\left(N_{L/k}:R_{L/k}\mathbb G_m\longrightarrow\mathbb G_m\right).
$$
Here $R_{L/k}$ 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 $(z_1,\ldots,z_d)\mapsto z_1\cdots z_d$. 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.
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