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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)

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2 Answers 2

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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 $K$.

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    $\begingroup$ Why mention fiber functors? Since you mention the equivalence of categories (which is proved over fields in more elementary references, such as Borel's book), and split torus corresponds to trivial Galois action, the question is the same as for "splitting fields" of discrete Galois modules finite free over $\mathbf{Z}$. The same argument as for finite discrete Galois modules or finite discrete Galois sets (or etale torsion on abelian varieties) then applies: the kernel of the action map from the Galois group is an open subgroup whose corresponding field is Galois over $k$ and does the job. $\endgroup$
    – Marguax
    Commented Sep 21, 2013 at 15:36
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    $\begingroup$ Marguax, I am convinced that your answer works, but how do you justify that it really does the job. For instance, if $L$ is a totally inseparable of $k$ extension, how do you see that $T$ does not split on $L$? $\endgroup$
    – Joël
    Commented Sep 21, 2013 at 17:10
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    $\begingroup$ @Joel: The equivalence of tori and Galois lattices uses that tori over separably closed fields are split (all in Borel). Since a purely inseparable field extension induces an isomorphism of Galois groups, we win. Or: if $L/k$ is purely inseparable finite and $T, T'$ are $k$-tori and $f:T'_L \simeq T_L$ is an isomorphism then $p_1^{\ast}(f), p_2^{\ast}(f)$ are isomorphisms over $L \otimes_k L$ which coincide due to equality on the relatively schematically dense collection of finite etale torsion-levels (as may be checked on the diagonal $L$-point!), so $f$ descends to $k$ by fppf descent. $\endgroup$
    – Marguax
    Commented Sep 21, 2013 at 17:31
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    $\begingroup$ @Joel: Another "fancy" way to think about it is the fact that the automorphism scheme of a split torus is etale (though one only needs that the automprphism functor is formally unramified, which comes down to exactly the same argument as in my preceding comment, up to how general you want the base ring to be). The bare-hands argument with descent theory applies to any $L$-homomorphism, and the reasoning applies to abelian varieties as well (known to Chow): if $A, A'$ are abelian varieties over $k$ and $L/k$ is purely inseparable, any $L$-homomorphism $A'_L \rightarrow A_L$ descends to $k$. $\endgroup$
    – Marguax
    Commented Sep 21, 2013 at 17:33
  • $\begingroup$ Marguax: Good, I like that. $\endgroup$
    – Joël
    Commented Sep 21, 2013 at 18:47
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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.

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    $\begingroup$ The proof of existence of a finite separable splitting field in Borel's textbook (Prop. 8.11 in Ch. III) seems to be a bit simpler than the one in Prop. 1.5 of the IHES paper. $\endgroup$
    – Marguax
    Commented Sep 21, 2013 at 20:22
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    $\begingroup$ @Marguax: True. I wanted to add an online reference and also go back to the modern origins of rationality discussions. (Note that Borel's lectures at Columbia in the late 1960s were written up by Bass, who lectured on this at Bowdoin in summer 1968; this part of Borel's lectures was unchanged in the expanded GTM edition.) The treatment in scheme language evolved through SGA3 and Demazure-Gabriel to the notes of Milne and the recent book by Conrad-Gabber-Prasad. $\endgroup$
    – Jim Humphreys
    Commented Sep 22, 2013 at 13:25
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    $\begingroup$ working link $\endgroup$
    – Kenny Lau
    Commented Jun 5, 2018 at 22:38

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