The question is whether every field extension $L/K$ of degree $2018$ is primitive; i.e. $K\subset L$ field extension and $[L:K]=2018$ implies $L=K(\alpha)$ for some $\alpha\in L$, true or false. The prime factorisation is $2\cdot1009=2018$. I know at least two theorem that could help me here: the theorem of the primitive element and another lemma about intermediate fields.
If $K\subset L$ is a finite separable extension (hence also algebraic), then there exists an $\alpha\in L$ such that $L=K(\alpha)$.
Let $K\subset L$ be a finite field extension. Then equivalent are: [$L$ is primitive]$\iff$[there are only finitely many intermediate fields $E$ with $K\subset E\subset L$].
I don't now which one of these two will be easiest to show and I wouldn't know how to do either one of them. Any suggestions? Thanks in advance.