Definition. finitely generated, simple extensions and primitive elements [1304]

  1. A field extension $K / F$ is called finitely generated if there exist $\alpha_{1}, \dots, \alpha_{n} \in K$ such that $K = F(\alpha_{1}, \dots, \alpha_{n})$.
  2. A field extension $K / F$ is called simple if $K = F(\alpha)$ for some $\alpha \in K$. In this case, $\alpha$ is called a primitive element for the field extension $K / F$.