Classification of Real Tori
Take \(k = \mathbb{R}\), so \(k_s = \mathbb{C}\) and \(\Theta = \operatorname{Gal}(\mathbb{C}/\mathbb{R}) = C_2 = \{1, \tau\}\). By F-Tori Are Classified by Galois Modules, real tori correspond to \(\mathbb{Z}[C_2]\)-lattices.
Lattice input. Every \(\mathbb{Z}[C_2]\)-lattice is a direct sum, unique up to reordering, of the three indecomposables: \[ \mathbb{Z}\;(\tau = +1), \qquad \mathbb{Z}^-\;(\tau = -1), \qquad \mathbb{Z}[C_2]\;(\text{regular, } \tau \text{ swaps the basis}). \] (Integral representation theory of the cyclic group of order 2; uniqueness is the Krull–Schmidt property for this order.)
Torus output. Under the classification the three indecomposables correspond to:
| lattice | torus | real points |
|---|---|---|
| \(\mathbb{Z}\) (trivial) | \(\mathbb{G}_{m,\mathbb{R}}\) (Split Torus) | \(\mathbb{R}^\times\) |
| \(\mathbb{Z}^-\) (sign) | \(S^1 = T^1_{\mathbb{C}/\mathbb{R}}\) (Norm-One Torus of a Separable Extension) | \(\{z \in \mathbb{C}^\times : z\bar z = 1\}\) |
| \(\mathbb{Z}[C_2]\) (regular) | \(\operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m\) (Weil Restriction of Tori) | \(\mathbb{C}^\times\) |
Hence every real torus is, uniquely up to reordering, \[ T \;\cong\; \mathbb{G}_m^{\,a} \times (S^1)^{\,b} \times \bigl(\operatorname{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_m\bigr)^{c}, \qquad a, b, c \geq 0 . \] The rank is \(a + b + 2c\), the split rank is \(a + c\) (the \(\tau = +1\) fixed sublattice of \(X_*\) has rank \(a+c\)), and \(T\) is anisotropic iff \(a = c = 0\), i.e. \(T \cong (S^1)^b\) — the compact real tori.