Tensor product of fields

Summary

Let $E$ and $F$ be extensions of a field $K$. Their tensor product $E \otimes_K F$ is the tensor product of vector spaces over $K$, with the product $(a \otimes b)(c \otimes d) = ac \otimes bd$. It is usually not a field. Each way to put $E$ and $F$ inside one field gives a quotient of $E \otimes_K F$ by a prime ideal. The field of fractions of this quotient is the compositum, so the tensor product records all these ways at once. For a finite separable extension $E = K(\alpha)$ it is a product of fields, one for each irreducible factor over $F$ of the minimal polynomial of $\alpha$.