Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 18H / a

The extension $L/K$ is finite when $L$ is finite-dimensional as a vector space over $K$, and its degree is $[L:K]={\dim }_{K}L$.

The extension $L/K$ is separable when it is algebraic and the minimal polynomial over $K$ of every element of $L$ has no repeated root in an algebraic closure.

The extension $L/K$ is simple when there is an element $\alpha \in L$ such that $L=K(\alpha )$.

Finite and separable imply simple by the primitive element theorem. Separable and simple imply finite: if $L=K(\alpha )$ and the extension is separable, then $\alpha$ is algebraic, so $[L:K]=\mathrm{deg}{m}_{\alpha ,K}<\infty$.

Finite and simple do not imply separable. In characteristic $p$, takeThen $L/K$ has degree $p$ and is simple, but the generator has inseparable minimal polynomial $X^p-t$. These are the finite separable simple extension implications.