Field embedding

Definition. A field embedding is an injective homomorphism of fields $\varphi:K\to L$ (preserving $1$). Equivalently, it is a ring monomorphism between fields, hence also an injective function .

Any field extension $L/K$ can be presented as an embedding $K\hookrightarrow L$; after identifying $K$ with its image, we write $K\subseteq L$.

See also. field automorphism , simple extension .

Examples.

1. The inclusion $\mathbb{Q}\hookrightarrow \mathbb{R}$ is a field embedding.
2. For $L=\mathbb{Q}(\sqrt2)$, there are two embeddings $L\hookrightarrow \mathbb{C}$ fixing $\mathbb{Q}$: $\sqrt2\mapsto \sqrt2$ and $\sqrt2\mapsto -\sqrt2$.
3. The map $\mathbb{F}_p \hookrightarrow \mathbb{F}_{p^n}$ is a field embedding for every $n\ge 1$.