Unramified Torus

Let \(k\) be a Henselian discretely valued field (see Local Fields), \(K \subseteq k_s\) its maximal unramified extension, and \(I = \operatorname{Gal}(k_s/K) \subseteq \Theta\) the inertia subgroup.

A torus \(T\) over \(k\) is unramified if \(T_K\) is split. Equivalently:

• the inertia group \(I\) acts trivially on \(X^*(T)\); equivalently
• \(T\) splits over a finite unramified extension of \(k\) (by The Minimal Splitting Field of a Torus the minimal splitting field of an unramified torus is a finite subextension of \(K/k\)).

For an unramified torus the Galois-lattice data reduces to an action of the unramified quotient \(\operatorname{Gal}(K/k)\) on \(X^*(T)\); over a finite or, more generally, quasi-finite residue field this quotient is (topologically generated by) Frobenius, so unramified tori are classified by a single automorphism of the lattice up to conjugacy.