Corollary

There exist composite extensions of E and F.

For since the commutative ring $E \otimes_K F$ is not reduced to 0, it has prime ideals : Krull's theorem (I, p. 104) proves the existence of maximal ideals and every maximal ideal is prime.

We can make this corollary more precise as follows. Let $(E, u)$ and $(F, v)$ be two extensions of K ; choose a maximal ideal m of the commutative ring $E \otimes_K F$ and put $L = (E \otimes_K F)/m$; then L is an extension of K. For $x \in E$ write $u'(x)$ for the residue class of $x \otimes 1$ mod m and similarly put $v'(y)$ for the residue class of $1 \otimes y$ mod $m$ for all $y \in F$. We then have a commutative diagram of field homomorphisms

By replacing $(L, u')$ by an isomorphic extension of E we may suppose that L contains E as subfield and that $u'$ is the canonical injection of E in L. By changing notation we thus obtain the following scholium :