I have the two simple extensions $F \subseteq F(\theta)$ and $F \subseteq F(\gamma)$, which are stated to be Galois extensions. We also have char$(F) = 0$. The problem is whether or not $F \subseteq F(\theta, \gamma)$ is a Galois extension.
Let $E = F(\theta,\gamma)$ for simplicity. It's clear that $E$ is a finite extension of $F$. Now let $\{f_{i}(x)\}$ be the familiy of polynomials in $F[x]$ of which $F(\theta)$ is the splitting field. Then $E$ is also a splitting field of that family of polynomials, and so $E$ is a normal extension of $F$. The part I cannot quite figure out is whether or not $E$ is a separable extension of $F$. I have a gut feeling that it should be, but gut feelings doesn't really accomplish much in mathematics.
We have $F(\theta, \gamma) = \{a + b\theta + c\gamma + d\theta\gamma : a,b,c,d \in F\}$. Given that both $F(\theta)$ and $F(\gamma)$ are separable extensions of $F$, any element in $E$ of the form $a + b\theta + c\gamma$ is separable over $F$ (is this correct?). This essentially leaves the elements including $d\theta\gamma$ for inspection, and this is where I'm stuck. I tried to look at the product of the minimal polynomials of $\theta$ and $\gamma$ to see if $d\theta\gamma$ was separable over $F$, but I became uncertain whether or not this would even imply anything meaningful.
I'm looking for any kind of hints, vague or not, to how I can approach this problem. Also, if I have made intermediate mistakes, it would be fantastic if someone could point them out.
EDIT: The extension $E$ is the splitting field of $p(x)q(x)$ where $p$ and $q$ are the minimal polynomials of $\theta$ and $\gamma$ respectively. Apparently I missed the results in my book linking separability of an algebraic extension and the fact that the ground field has char $0$, making that part of the problem trivial.