 Field extension

A field extension $L/K$ consists of a field $L$ containing a specified subfield $K$. Its degree $[L:K]$ is the dimension of $L$ as a vector space over $K$, possibly infinite. The inclusion specifies which scalars are available; enlarging them can supply missing roots of a polynomial.^[ref]

For example, $\mathbb{C}/\mathbb{R}$ has basis $1,i$ and degree two, while $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ has basis $1,\sqrt{2}$. The field extension $\mathbb{Q}(t)/\mathbb{Q}$ with $t$ an indeterminate is not algebraic: no nonzero rational-coefficient polynomial vanishes at $t$. The tower law multiplies degrees along finite towers $K\subseteq L\subseteq M$.