A field extension of degree 8

Question

I would really appreciate it if you give me a hint on the following question:

If $K \subset F$ is a field extension of degree 8, then we must have $F=K(a,b,c)$ for some a, b and c in F.

Answer 1

If $a\in F\setminus K$, then $K(a)$ is of index at least two over $K$. If it is actually already F, just pick b,c at random. Otherwise, Picking $b\in F\setminus K(a)$ makes $K(a,b)$ of degree at least 4.

The last steps are exactly the same: can you carry it out without me spelling it out?

The easy generalization is that an extension of degree having n prime factors can be bridged with at most n adjunctions.