Field extension

Definition. A field extension $L/K$ consists of fields $K$ and $L$ together with an injective field homomorphism $\iota:K\hookrightarrow L$. Identifying $K$ with $\iota(K)$, one usually writes $K\subseteq L$ and says “$L$ is an extension field of $K$”.

When $K\subseteq L$, the field $L$ is naturally a vector space over $K$. If $\dim_K L$ is finite, it is the degree of the extension $[L:K]$.

See also. field embedding , intermediate field , tower of fields .

Examples.

1. $\mathbb{R}\subseteq \mathbb{C}$ is a field extension; in fact $[\mathbb{C}:\mathbb{R}]=2$.
2. $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt{2})$ is a field extension obtained by adjoining $\sqrt2$.
3. $\mathbb{F}_p\subseteq \mathbb{F}_{p^n}$ is a finite field extension (see finite fields ).