** Reformulation of the question (see below for the original question):** Let $K$ be an algebraic number field and $D$ a finite-dimensional $K$-division algebra. Is there a description of the field extensions $L\supseteq K$ such that $L\otimes_K D$ is a division algebra?

I have encountered this question in the context of representation theory of finite groups, see below.

*Original question:*

I have asked this question at math.stackexchange, but have not received an answer so far. Also, I'm not entirely sure that this is a suitable question for mathoverflow... See this link for the original question.

Let $K$ be an algebraic number field, $G$ a finite group and $V$ an irreducible $KG$-module with character $\chi$. Since V is irreducible, $\chi$ is of the form $\chi=m(\vartheta_1+...+\vartheta_s)$, where the $\vartheta_i$ are the distinct conjugates of an absolutely irreducible character of G under the Galois group $\mathrm{Gal}(K[\vartheta]/K)$ of the character field $K[\vartheta]:=K[\vartheta(g) ~:~ g \in G]$ and m is the Schur index. We have $s=[K[\vartheta]:K]$.

My question is: Is there a characterisation of field extensions $L\supseteq K$ such that $V_L := L\otimes_K V$ is a reducible $LG$-module?

Since I asked this question at stackexchange I have found that for $m=1$ (and therefore $s\geq 2$, because otherwise we already have an absolutely irreducible representation) we have that $V_L$ is reducible if and only if $L$ contains an intermediate field $K\subseteq F \subseteq K[\vartheta]$ such that $\mathrm{Gal}(K[\vartheta]/F)\leq \mathrm{Gal}(K[\vartheta]/K)$ has more than one orbit on $\{\vartheta_1,...,\vartheta_s\}$.

However, I am unsure about the case of $m\geq 2$, which I assume is more subtle. Is there a similarly succinct characterisation of field extensions $L|K$ such that $V_L$ is reducible?

*Edit:* As Geoff Robinson points out, this problem may be restated as, "Which fields $L\supseteq K$ have the property that $\mathrm{End}_{LG}(V_L)$ is not a division algebra?" which is basically a question about division algebras over number fields.

Considering the (small) example of the quaternion group $Q_8$ over the rationals and its irreducible four-dimensional representation $V$, a quick computation shows that $\mathrm{End}_{\mathbb{Q}Q_8}(V)$ becomes a matrix ring over $\mathbb{Q}(\sqrt{-1})$, $\mathbb{Q}(\sqrt{-2})$, $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{-5})$ but not over $\mathbb{Q}(\sqrt{-7})$.

I have, at present, no idea how to characterize these fields, especially in terms of some group theoretic or character theoretic property of $Q_8$. Then again, this is probably not to be exprected, considering that $Q_8$ and $D_8$ have "equal" character tables, but the two-dimensional representations have distinct Schur indices...