Tamely Ramified Torus

Let \(k\) be a Henselian discretely valued field with residue characteristic \(p \geq 0\), and let \(P \subseteq I \subseteq \Theta\) denote the wild inertia subgroup (the pro-\(p\) radical of the inertia group \(I\); trivial when \(p = 0\)).

A torus \(T\) over \(k\) is tamely ramified if it splits over a tamely ramified Galois extension of \(k\). Equivalently, the wild inertia group \(P\) acts trivially on \(X^*(T)\).

Special cases:

• every unramified torus is tamely ramified;
• if \(p = 0\), every torus over \(k\) is tamely ramified;
• if \(p > 0\), a torus whose minimal splitting field (The Minimal Splitting Field of a Torus) has degree prime to \(p\) is tamely ramified.