[[Field theory MOC]]
# Intermediate field extension

Suppose $F : L : K$ is a tower of [[Field extension|field extensions]]
$$
\begin{matrix}
F  \\
| \\
L \\
| \\
K
\end{matrix}
$$
Then in particular one has a [[Vector space over a field extension]], whence
$$
\begin{align*}
[F:K] = [F:L][L:K]
\end{align*}
$$
and $F:K$ is finite iff both $F:L$ and $L:K$ are. #m/thm/field

## Corollaries of this corollary

1. Given a tower $F:L:K$, it follows both $[F:K]$ and $[F:L]$ divide $[F:K]$. ^C1

#
---
#state/tidy | #lang/en | #SemBr