Compositum of fields

Summary

The compositum of two subfields $E$ and $F$ of a field $\Omega$ is the smallest subfield of $\Omega$ that contains both, written $EF$. It is the field generated by $E \cup F$, and when $E = K(\alpha_1, \ldots, \alpha_n)$ it equals $F(\alpha_1, \ldots, \alpha_n)$. Composita appear throughout Galois theory: the compositum of two Galois extensions is Galois, and its degree is controlled by linear disjointness. Without a common field $\Omega$ the compositum is not defined, and the tensor product of fields describes all possible ways to form one.