Splitting field of a Torus Let $T$ be a torus over a non-necessary perfect field $k$. Let $\bar k$ an algebraic closure of $k$. Is there a smallest extension $k'$ of $k$ in $\bar k$ such that $T \times_{{\rm spec}\, k} {\rm spec}\, k'$ is split over $k'$?
(I have asked this on stack exchange, but did not receive a satisfactory answer)
 A: I think so: it is the Galois subextension $K/k$ of $\overline{k}/k$ cut out by the Galois representation on the character group of $T$ (I will freely use the anti-equivalence between tori and integral representations stated precisely, e.g., in B.3.6 of http://math.stanford.edu/~conrad/papers/luminysga3.pdf ). This $K$ certainly splits $T$. On the other hand, if a subextension $L/k$ splits $T$, then, looking at $\pi_1(L, \overline{k}) \rightarrow \pi_1(k, \overline{k})$, every automorphism of the fiber functor of $L$ must induce the trivial automorphism on the pullback to $L$ of the Galois cover $K/k$. In other words, this pullback splits completely, meaning that $L$ contains the roots of a polynomial defining $K/k$ in $\overline{k}/k$, i.e., $L$ contains $K$.
A: As the surroounding discussion of the "fancy" answer by Kestutis makes clear, this kind of splitting field question can be approached in somewhat different ways depending on the framework used.  It's probably useful to recall the straightforward early approach of Borel and Tits in their 1965 IHES paper here.  They make the basic move in their first section toward recasting the problem in terms of Galois actions.   
It's clear from their formulation that you get a unique smallest finite Galois subextension inside a given separable closure of $k$ which splits the torus.   It's also clear that this can be restated in scheme language, for which a reference might be the book by Demazure-Gabriel.  The basic question itself is not so hard to answer, but the style of the answer will definitely vary with the language used and the applications one has in mind.
