Degree of a field extension

Let $E/F$ be a field extension . The degree of $E/F$, denoted $[E:F]$, is the dimension of $E$ as a vector space over $F$.

Concretely, a subset $B\subseteq E$ is an $F$-basis of $E$ if every $x\in E$ can be written uniquely as a finite sum

The cardinality of a basis is independent of the choice of basis; this cardinal is $[E:F]$. The extension is called finite if $[E:F]<\infty$.

If $F\subseteq K\subseteq E$ is a tower of fields and the degrees are finite, then the tower law states

Examples

1. $[\mathbb{Q}(\sqrt2):\mathbb{Q}]=2$, with basis $\{1,\sqrt2\}$ over $\mathbb{Q}$.
2. $[\mathbb{C}:\mathbb{R}]=2$, with basis $\{1,i\}$ over $\mathbb{R}$.
3. For a prime $p$ and $n\ge 1$, $[\mathbb{F}_{p^n}:\mathbb{F}_p]=n$. (In particular, $\mathbb{F}_{p^n}/\mathbb{F}_p$ is finite of degree $n$.)