Linearly disjoint extensions

Summary

Let $E$ and $F$ be extensions of a field $K$ inside a common field. They are linearly disjoint over $K$ when elements of $E$ that are linearly independent over $K$ stay independent over $F$. Equivalently, the natural map from the tensor product $E \otimes_K F$ to the common field is injective. For finite extensions this means that the degree of the compositum is as large as possible: $[EF:F] = [E:K]$. The notion is the field version of independence, and it is used to compute degrees and Galois groups of composita.