Weil Restriction of Tori
Let \(E/k\) be a finite separable field extension, \(\Theta_E = \operatorname{Gal}(k_s/E) \subseteq \Theta\), and let \(S\) be a torus over \(E\) of rank \(m\). The Weil restriction \(R_{E/k}S\) — the affine \(k\)-group with functor of points \[ (R_{E/k}S)(A) \;=\; S(A \otimes_k E) \quad \text{for } k\text{-algebras } A \] — is a torus over \(k\) of rank \([E:k]\,m\), with \[ X^*\!\bigl(R_{E/k}S\bigr) \;=\; \operatorname{Ind}_{\Theta_E}^{\Theta} X^*(S), \qquad X_*\!\bigl(R_{E/k}S\bigr) \;=\; \operatorname{Ind}_{\Theta_E}^{\Theta} X_*(S) \] as \(\Theta\)-modules. In particular \(X^*(R_{E/k}\mathbb{G}_m) = \mathbb{Z}[\Theta/\Theta_E]\), the permutation module on the coset space, and \((R_{E/k}\mathbb{G}_m)(k) = E^\times\).
Proof
Representability of \(R_{E/k}S\) by an affine \(k\)-group of finite type is the standard existence of Weil restriction along the finite locally free morphism \(\operatorname{Spec} E \to \operatorname{Spec} k\) (source span). To see that it is a torus, base change to \(k_s\): since \(E/k\) is separable, \(E \otimes_k k_s \cong \prod_{\iota} k_s\) indexed by the \([E:k]\) embeddings \(\iota : E \to k_s\), giving \[ (R_{E/k}S)_{k_s} \;\cong\; \prod_{\iota} S \times_{E, \iota} k_s \;\cong\; \mathbb{G}_{m}^{[E:k]\,m}, \] so \(R_{E/k}S\) is a torus of the stated rank.
The displayed product decomposition identifies \(X^*((R_{E/k}S)_{k_s})\) with \(\bigoplus_{\iota} X^*(S \times_{E,\iota} k_s)\); the group \(\Theta\) permutes the factors through its action on the embeddings \(\iota\), i.e. on \(\Theta/\Theta_E\), and \(\Theta_E\) acts on the factor at the distinguished embedding through its action on \(X^*(S)\). This is exactly the induced module \(\operatorname{Ind}_{\Theta_E}^{\Theta} X^*(S)\), and dually for cocharacters. Finally \((R_{E/k}S)(k) = S(k \otimes_k E) = S(E)\), in particular \(E^\times\) for \(S = \mathbb{G}_m\).