Degree of a field extension

Remarks

The degree is a positive integer or an infinite cardinal. One has $[L : K] = 1$ if and only if $L = K$. Examples: $[\mathbb{C} : \mathbb{R}] = 2$, with the basis $1, i$. Also, $[\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2$, with the basis $1, \sqrt{2}$. The field $\mathbb{R}$ has infinite degree over $\mathbb{Q}$, because $\mathbb{Q}$ is countable and $\mathbb{R}$ is not.