Degree of a field extension

Definition. Let $L/K$ be a field extension . The degree of the extension is

the dimension of $L$ as a vector space over $K$.
If this dimension is finite, $L/K$ is a finite extension; otherwise $[L:K]=\infty$.

In a tower $K\subseteq F\subseteq L$ with finite degrees, the tower law says $[L:K]=[L:F]\,[F:K]$.

See also. tower of fields , simple extension .

Examples.

1. $[\mathbb{C}:\mathbb{R}]=2$ with basis $\{1,i\}$.
2. $[\mathbb{Q}(\sqrt2):\mathbb{Q}]=2$ with basis $\{1,\sqrt2\}$.
3. If $q=p^n$, then $[\mathbb{F}_{q}:\mathbb{F}_p]=n$ (see finite fields ).