Field extension

Let $F$ and $E$ be fields . A field extension of $F$ is a pair $(E,\iota)$ where $\iota:F\hookrightarrow E$ is an injective field homomorphism. In practice one identifies $F$ with its image $\iota(F)\subseteq E$, writes simply $F\subseteq E$, and denotes the extension by $E/F$.

If $E/F$ is a field extension, any subfield $K$ with $F\subseteq K\subseteq E$ is an intermediate field . Such a $K$ produces a tower of fields $F\subseteq K\subseteq E$. When $E$ is finite-dimensional as an $F$-vector space, the degree $[E:F]$ is defined.

A map of extensions is typically expressed via a field embedding $E\hookrightarrow E'$ that restricts to the identity on $F$; a bijective embedding $E\to E$ fixing $F$ is a field automorphism over $F$.

Examples

1. $\mathbb{R}\subseteq \mathbb{C}$ is a field extension, often written $\mathbb{C}/\mathbb{R}$.
2. $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt{2})$ is a field extension obtained by adjoining $\sqrt{2}$.
3. For a prime $p$ and $n\ge 1$, $\mathbb{F}_p \subseteq \mathbb{F}_{p^n}$ is a finite field extension (where $\mathbb{F}_{p^n}$ is a finite field of size $p^n$).