Why $\overline{\mathbb Q}$ has no subfield index of $3$?

Question

Show $\overline{\mathbb Q}$ has no subfield index of $3$, where $\mathbb Q$ is the rational number field.

In fact, I'm wonder about that statement is true.

May I ask you any idea?

Answer 1

In the context of the infinite Galois extension $\overline{\Bbb Q}\mid \Bbb Q$ the Artin-Schreier Theorem is the natural result to use. Since the title suggests this context, it seems to be appropriate. We obtain that there is no finite index subfield of index $n>2$ - the only finite groups arising as absolute Galois groups are $1$ and $C_2$. On the other hand, every profinite group arises as a Galois group.