Theorem Classification of Rank-One Tori
theorem admitted

Classification of Rank-One Tori

Every torus of rank \(1\) over a field \(k\) is isomorphic to exactly one of:

  • the split torus \(\mathbb{G}_m\);
  • the norm-one torus \(R^1_{E/k}\mathbb{G}_m\) of a separable quadratic extension \(E/k\),

and \(R^1_{E/k}\mathbb{G}_m \cong R^1_{E'/k}\mathbb{G}_m\) if and only if \(E = E'\) inside \(k_s\). Thus rank-one tori are parametrized by separable quadratic extensions of \(k\) together with the split one.

Proof

By F-Tori Are Classified by Galois Modules a rank-one torus corresponds to a continuous action of \(\Theta\) on \(\mathbb{Z}\), i.e. a continuous homomorphism \(\chi : \Theta \to \operatorname{Aut}(\mathbb{Z}) = \{\pm 1\}\) (conjugation is trivial in an abelian target, so isomorphism classes are homomorphisms, not just conjugacy classes). If \(\chi\) is trivial the lattice is trivial and the torus is split. Otherwise \(\ker \chi\) is an open subgroup of index \(2\), i.e. \(\ker\chi = \operatorname{Gal}(k_s/E)\) for a uniquely determined separable quadratic extension \(E/k\), and the lattice is the sign lattice of \(E/k\) — which by Norm-One Torus of a Separable Extension is the lattice of \(R^1_{E/k}\mathbb{G}_m\). Distinct \(E\) give distinct kernels, hence non-isomorphic lattices.