Norm-One Torus of a Separable Extension

Let \(E/k\) be a finite separable extension. The norm map on points \(N_{E/k} : E^\times \to k^\times\) is algebraic: it is the map of tori \[ N \;:\; R_{E/k}\mathbb{G}_m \longrightarrow \mathbb{G}_m \] which on characters sends the generator of \(X^*(\mathbb{G}_m) = \mathbb{Z}\) to the norm element \(\sum_{c \in \Theta/\Theta_E} c \in \mathbb{Z}[\Theta/\Theta_E]\). The norm-one torus is its kernel: \[ R^{1}_{E/k}\mathbb{G}_m \;=\; \ker\bigl(N : R_{E/k}\mathbb{G}_m \to \mathbb{G}_m\bigr), \qquad R^{1}_{E/k}\mathbb{G}_m(k) = \{x \in E^\times : N_{E/k}(x) = 1\}. \]

It is a torus of rank \([E:k]-1\): the norm element spans a saturated \(\Theta\)-stable sublattice \(\mathbb{Z}\cdot N \subseteq \mathbb{Z}[\Theta/\Theta_E]\) (a generator with content one), so by Subtori and Quotients via the Character Lattice the kernel is a subtorus with \[ X^*\bigl(R^1_{E/k}\mathbb{G}_m\bigr) \;=\; \mathbb{Z}[\Theta/\Theta_E] \,/\, \mathbb{Z}\!\cdot\! N . \]

Quadratic case. For \([E:k] = 2\) with \(\operatorname{Gal}(E/k) = \{1,\sigma\}\): \(X^*(R^1_{E/k}\mathbb{G}_m) \cong \mathbb{Z}\) with \(\sigma\) acting by \(-1\) (the sign lattice). Its invariants vanish, so the quadratic norm-one torus is anisotropic — the smallest non-split torus, and the running counterexample of this topic: it witnesses the failure of the split-anisotropic decomposition to be direct, and, over discretely valued fields with ramified \(E/k\), the failure of the Iwahori subgroup to exhaust the maximal bounded subgroup (recorded at the arithmetic stage; source span).