Tower of fields

Definition. A tower of fields is a chain of inclusions

where each inclusion is a field extension . The middle field $F$ is an intermediate field of $L/K$.

If the degrees are finite, the tower law gives

See also. degree of an extension , separability in towers .

Examples.

1. $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt2)\subseteq \mathbb{Q}(\sqrt2,\sqrt3)$ is a tower with $[\,\mathbb{Q}(\sqrt2):\mathbb{Q}\,]=2$.
2. $\mathbb{F}_p \subseteq \mathbb{F}_{p^2} \subseteq \mathbb{F}_{p^6}$ is a tower of finite fields with degrees $2$ and $3$.
3. $\mathbb{Q}\subseteq \mathbb{Q}(t)\subseteq \mathbb{Q}(t,\sqrt{t})$ is a tower where the bottom extension is transcendental and the top over the middle is algebraic of degree $2$.