An old qual problem reads
Let $D$ be a 9-dimensional central division algebra over $\mathbb{Q}$ and $K \subset D$ be a field extension of $\mathbb{Q}$ of degree $>1$. Show that $K \otimes_\mathbb{Q} K$ is not a field and deduce that $D \otimes_\mathbb{Q} K$ is no longer a division algebra.
I'm not sure the appropriate tool to use to approach this. Clearly $K$ is of degree $3$. Can someone help out? Perhaps you could also add a reference where this material is covered in more generality.
By the primitive element theorem, we can write $K = \mathbb{Q}[\alpha]/f(\alpha)$ for some irreducible polynomial $f$. Then
$$K \otimes K \cong K[\alpha]/f(\alpha)$$
which is not a field since $K$, by construction, contains a root of $f$, hence $K[\alpha]/f(\alpha)$ is a product of at least two fields (one for every irreducible factor of $f$ over $K$), one of which is $K$. (If $K$ were Galois then we'd be able to say something stronger: namely, $K \otimes K$ would be a product of $\deg f$ copies of $K$.)
It follows that $D \otimes K$ contains a subring $K \otimes K$ which isn't a domain, so isn't a division ring. I'm a little worried here because I haven't used the hypothesis that $D$ is $9$-dimensional, but as far as I can see this isn't necessary.