---
tag: X4ZZKC319Q
kind: definition
name: Compositum of fields
msc: [12F05, 12F99]
defines: T1S29NN8PE
uses: [07QY5RPFBR, 0MA5B744K3]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_product_of_fields
---

## Definition

Let $\Omega$ be a field and let $E$ and $F$ be [[subfields|07QY5RPFBR]] of $\Omega$.
The *compositum* $EF$ of $E$ and $F$ is the intersection of all subfields of $\Omega$ that contain $E \cup F$.
More generally, the compositum of a family of subfields of $\Omega$ is the intersection of all subfields of $\Omega$ containing every member.

## 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|0MA5B744K3]].
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 $F$.