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Let $G$ be a simple group over a field $F$, and let $K$ be an extension of $F$ such that $G$ base changed to $K$ is split. Does this mean that $K$ splits a maximal torus of $G$?

Context: I want to know whether the following statement holds: $K$ is a quadratic extension of $F$ splitting a quaternion algebra $D$ over $F$, then $K$ embeds in $D$ over $F$.

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  • $\begingroup$ Nope! Indeed, a purely inseparable extension cannot ever split a torus (due to the "Galois lattice" viewpoint) but it sure can split such a group. For example, if $k$ is a local function field of characteristic $p>0$ and $G$ is the group of norm-1 units in a central division algebra of rank $n^2$ over $k$ then it is split by any extension of degree $n$ by local class field theory, so taking $n=p$ and the extension $k^{1/p}/k$ of degree $p$ gives a counterexample. Perhaps you meant to ask that $K/F$ is separable? Probably the answer remains negative, but let's see what you want. $\endgroup$
    – nfdc23
    Commented Oct 19, 2015 at 2:48
  • $\begingroup$ Also, it is a very classical fact (Theorem 4.12 in Jacobson's "Basic Algebra II") that a degree-$n$ extension of fields $K/F$ splits a degree-$n^2$ central simple algebra $A$ if and only if $K$ occurs as an $F$-subalgebra of $A$. (These give rise to rather special $F$-tori when $K/F$ is separable, by the way, and have nothing to do with $F$-tori otherwise.) So for the purpose of the motivation you don't need the general question for algebraic groups. Or is your point to understand the result for central simple algebras as a special case of a more general result? $\endgroup$
    – nfdc23
    Commented Oct 19, 2015 at 2:52
  • $\begingroup$ I'm interested in the question over number fields, so I'm happy to assume that my extensions are separable. Thanks for your answer - I worked a messy answer out for the quaternions, and it's good to have it confirmed for CSAs. $\endgroup$
    – user81663
    Commented Oct 19, 2015 at 3:24
  • $\begingroup$ I would be curious about the question in general, and now that you've given your counterexample in the function field case, I think something similar will work in general. For a division algebra of degree $n^2$, there should be $S_n$ number fields $K$ which split it, and those will give counterexamples - using the Galois action on lattices viewpoint, an extension will split a torus only if it contains a large enough (I'm being vague here, but you probably see what I mean) galois subextension. As $K$ is an $S_n$ field, it's not going to have this property. $\endgroup$
    – user81663
    Commented Oct 19, 2015 at 3:33
  • $\begingroup$ For any $n > 2$ and any non-Galois degree-$n$ finite separable extension $K$ of a global field $F$, class field theory provides a central division algebra $D$ over $F$ split by $K$. The $F$-group $G$ of norm-1 units in $D$ is split by $K/F$. Maximal $F$-tori in $G$ are in bijective correspondence with commutative subfields $E$ of $D$ that are finite separable of degree $n$ over $F$. But $E \otimes_F K$ is never a split $K$-algebra. Indeed, if it splits then $E$ embeds $K$ over $F$, so $E\simeq K$ over $F$, so $K\otimes_F K$ splits as a $K$-algebra, contradicting that $K/F$ is non-Galois. $\endgroup$
    – nfdc23
    Commented Oct 19, 2015 at 4:09

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