What is field extension?

A field extension $L/K$ is a field $L$ containing $K$ as a subfield; $L$ is then a $K$-vector space, and the degree $[L:K]$ is its dimension. The extension is finite if $[L:K] < \infty$. The characteristic of a field is the generator $\geq 0$ of the kernel of $\Z \to K$, $n \mapsto n\cdot 1$: it is $0$ or a prime $p$; correspondingly $K$ contains a smallest subfield (prime field) isomorphic to $\Q$ or to $\mathbb F_p = \Z/p\Z$.