Intermediate field

A field F with K ⊆ F ⊆ L inside a field extension L/K.

Intermediate field

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

K \subseteq F \subseteq L.

Equivalently, $F$ is a subfield of $L$ that contains $K$ .

Intermediate fields are the “levels” in a tower of fields and interact with degrees via the tower law when the degrees are finite.

See also. degree of an extension , 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.
In $\mathbb{R}\subseteq \mathbb{C}$ , there is no proper intermediate field: the only intermediate fields are $\mathbb{R}$ and $\mathbb{C}$ (since $[\mathbb{C}:\mathbb{R}]=2$ ).
If $L=\mathbb{F}_{p^6}$ and $K=\mathbb{F}_p$ , then $\mathbb{F}_{p^2}$ and $\mathbb{F}_{p^3}$ are intermediate fields (unique of those sizes).