Intermediate field

Let $E/F$ be a field extension . An intermediate field (or subextension) of $E/F$ is a field $K$ such that

where both inclusions are inclusions of fields. Equivalently, $K$ is a subfield of $E$ that contains $F$.

Any intermediate field $K$ determines a tower

When the relevant degrees are finite, the tower law relates $[E:F]$, $[E:K]$, and $[K:F]$. In the special case of a Galois extension , intermediate fields are organized by the Galois correspondence .

Examples

1. In $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt2,\sqrt3)$, the fields $\mathbb{Q}(\sqrt2)$, $\mathbb{Q}(\sqrt3)$, and $\mathbb{Q}(\sqrt6)$ are intermediate fields between $\mathbb{Q}$ and $\mathbb{Q}(\sqrt2,\sqrt3)$.
2. If $m\mid n$, then $\mathbb{F}_{p^m}$ is an intermediate field of $\mathbb{F}_{p^n}/\mathbb{F}_p$; concretely, $\mathbb{F}_p \subseteq \mathbb{F}_{p^m} \subseteq \mathbb{F}_{p^n}.$
3. In $\mathbb{Q}\subseteq \mathbb{Q}(t)$ (the rational function field in one indeterminate), the subfield $\mathbb{Q}(t^2)$ is intermediate: $\mathbb{Q} \subseteq \mathbb{Q}(t^2) \subseteq \mathbb{Q}(t).$