Intermediate field

Definition

Let $L/K$ be a field extension. An intermediate field of $L/K$ is a subfield $E$ of $L$ with $K \subseteq E$.

Remarks

Then $L/E$ and $E/K$ are field extensions, and one calls $K \subseteq E \subseteq L$ a tower of fields. The fields $K$ and $L$ are intermediate fields of $L/K$. For example, $\mathbb{Q}(\sqrt{2})$ is an intermediate field of $\mathbb{R}/\mathbb{Q}$. The fundamental theorem of Galois theory describes the intermediate fields of a Galois extension.

Explanations

L1 human

An intermediate field sits between the small field and the big field. Galois theory gives a way to list all such fields, by a correspondence with subgroups of a group.