Compositum of fields

Remarks

The compositum $EF$ is the smallest subfield of $\Omega$ containing $E$ and $F$, so $EF = E(F) = F(E)$ in the notation of fields generated by adjoining elements. Its elements are the quotients $a/b$ with $b \ne 0$, where $a$ and $b$ are finite sums of products $xy$ with $x \in E$ and $y \in F$. If $K \subseteq E \cap F$ and $E/K$ is finite, then $[EF:F] \le [E:K]$.

Example: in $\Omega = \C$, the compositum of $E = \Q(\sqrt{2})$ and $F = \Q(\sqrt{3})$ is $\Q(\sqrt{2}, \sqrt{3})$. Check: this field contains $E$ and $F$, and any subfield of $\C$ containing $\sqrt{2}$ and $\sqrt{3}$ contains $\Q(\sqrt{2}, \sqrt{3})$.

Non-example: the union $E \cup F$ is not the compositum, because it is not a field. Indeed $\sqrt{2} + \sqrt{3}$ lies in neither $E$ nor $F$.