This question may have risen already, but I didn't find a clear answer. So I was learning about linear disjoint extensions. Let's set the stage:
I'm considering a base field $k$ and two extensions $L$ and $K$ embedded into a big extension $\Omega$: \begin{array}{ccccc} & \Omega & \\ / & & \backslash & \\ L & & K\\ \backslash & & /\\ & k & \\ \end{array} Now I know that $L/k$ and $K/k$ are said to be linearly disjoint if $$\begin{aligned} T:\ L\otimes_k K & \longrightarrow LK\\ l\otimes k & \longmapsto lk\\ \end{aligned}$$ is injective. Now this map intrigues me, for two reasons.
I heard someone say once that $L/k$ and $K/k$ are linearly disjoint iff $L\otimes_k K$ is a field. The implication $\Longleftarrow$ is clearly obvious, but is the reciprocal true in general?
I already saw in some notes that $T$ is surjective. Now I tried to wrap my head around this, and sampled some cases. The most generalcase I could find is if one of the extensions $L/k,\ K/k$ were algebraic, but I couldn't figure out the general case.
Any help would be appreciated.