Tower of fields - Knowlpedia

A tower of fields (or tower of extensions) is a chain of inclusions of fields

F \subseteq K \subseteq E.

Equivalently, it is a field extension $E/F$ together with an intermediate field $K$ between them. One often abbreviates this situation by writing $E/K/F$ .

Remarks

If the degrees are finite, towers are governed by the tower law:

[E:F]=[E:K]\,[K:F].

This allows one to compute or bound $[E:F]$ by passing through simpler intermediate steps.

Examples

$\mathbb{Q}\subseteq \mathbb{Q}(\sqrt2)\subseteq \mathbb{Q}(\sqrt2,\sqrt3)$ is a tower obtained by adjoining $\sqrt2$ first, then $\sqrt3$ .
If $m\mid n$ , then $\mathbb{F}_p\subseteq \mathbb{F}_{p^m}\subseteq \mathbb{F}_{p^n}$ is a tower of finite fields.
With $t$ an indeterminate, $\mathbb{Q}\subseteq \mathbb{Q}(t^2)\subseteq \mathbb{Q}(t)$ is a tower in which the top extension is transcendental, but $\mathbb{Q}(t)/\mathbb{Q}(t^2)$ has finite degree $2$ .