Intermediate field - Imported Knowlpedia Content

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

F \subseteq K \subseteq E,

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

Remarks

Any intermediate field $K$ determines a tower

F \subseteq K \subseteq E.

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

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