Theorem The Minimal Splitting Field of a Torus
theorem admitted

The Minimal Splitting Field of a Torus

Let \(T\) be a torus of rank \(n\) over \(k\). Then:

  1. \(T\) splits over a finite separable extension of \(k\);
  2. there is a unique minimal splitting field \(E/k\) inside \(k_s\); it is finite Galois over \(k\), namely the fixed field of \(\ker\bigl(\Theta \to \operatorname{Aut}(X^*(T))\bigr)\);
  3. the action of \(\operatorname{Gal}(E/k)\) on \(X^*(T)\) is faithful, giving an embedding \(\operatorname{Gal}(E/k) \hookrightarrow \operatorname{Aut}(X^*(T)) \cong \operatorname{GL}_n(\mathbb{Z})\).

In particular the isomorphism classes of rank-\(n\) tori over \(k\) correspond to conjugacy classes of continuous homomorphisms \(\Theta \to \operatorname{GL}_n(\mathbb{Z})\), and every torus is split by a finite Galois extension whose group is a finite subgroup of \(\operatorname{GL}_n(\mathbb{Z})\).

Proof

The \(\Theta\)-action on the discrete finitely generated group \(X^*(T)\) is continuous: each character of \(T_{k_s}\) is defined over a finite subextension, and a finite generating set is therefore stabilized by an open subgroup of \(\Theta\). Hence \(N = \ker(\Theta \to \operatorname{Aut} X^*(T))\) is an open normal subgroup, and its fixed field \(E\) is finite Galois over \(k\) with \(\operatorname{Gal}(E/k) = \Theta/N\) acting faithfully on \(X^*(T)\).

Over \(E\) the Galois group \(\operatorname{Gal}(k_s/E) = N\) acts trivially on \(X^*(T)\), so by F-Tori Are Classified by Galois Modules the torus \(T_E\) corresponds to the trivial \(\operatorname{Gal}(k_s/E)\)-lattice \(\mathbb{Z}^n\), i.e. \(T_E \cong \mathbb{G}_{m,E}^n\) is split. Any splitting field \(L \subseteq k_s\) of \(T\) must satisfy \(\operatorname{Gal}(k_s/L) \subseteq N\), i.e. \(L \supseteq E\), so \(E\) is the unique minimal one. The final classification statement is the anti-equivalence again.

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