How many fields are there strictly between $\Bbb Q(\zeta)$ and $\Bbb Q(\zeta^3)$

Question

Let $\zeta$ denotes the $12$th primitive root of unity. I want to know that how many fields are there strictly between $\Bbb Q(\zeta)$ and $\Bbb Q(\zeta^3)$ and what are they.

I think an obvious one is $\Bbb Q(\zeta^3+\zeta^{-3})$. But the solution says:

We have $\mathbb{Q}[\zeta] = \mathbb{Q}[i, \zeta']$ where $\zeta'$ is a primitive cube root of $1$ and $\pm i = \zeta^3$, etc...

I cannot see some explicit connection between the question to its solution... so what does Milne mean here? Could someone please point it out? Thanks!

Answer 1

$\Bbb Q(\zeta^3+\zeta^{-3})$ is a real subfield of $\mathbb{Q}(\zeta)$, hence cannot lie between $\Bbb Q(\zeta)$ and $\Bbb Q(\zeta^3)$. How is Milne's solution continuing? We have $\phi(12)=4$, hence $[\mathbb{Q}(\zeta):\mathbb{Q}]=4$, and $[\mathbb{Q}(\zeta^3):\mathbb{Q}]=[\mathbb{Q}(i):\mathbb{Q}]=2$. Now we are done.