I was doing some study on the programming language GAP and I came to know from here (in the very fist line) that " $\mathbb Q(\sqrt{5})$ is a number field that is not cyclotomic but contained in the cyclotomic field $\mathbb {Q}_5 = \mathbb Q(e^{\frac{2\pi i}{5}})$".
So I think it is an example that says that in general not all subfields of a cyclotomic field are cyclotomic. But a question came across in my mind from here, that I want to ask.
The above example deals with the $5$-th root of unity and $5$ is an odd prime. But I was thinking if we take $\mathbb Q(\theta)$ where $\theta$ is a primitive $2^n$-th root of unity for some $n>1$, will then same statement holds? In other words, I want to know
Are all subfields of $\mathbb Q(\theta)$ cyclotomic where $\theta$ is a primitive $2^n$-th root of unity for some $n>1$ ?
I am not at all good in algebraic number theory, so I am really sorry that I can not show much work from my side. I might be completely wrong or missing something trivial. Sorry again.
I will be really grateful if someone helps me to find an answer to this.
Thanks in advance.
Best Answer
It's certainly not the case that a subfield of $\Bbb Q(\theta)$ where $\theta=\exp(2\pi i/2^n)$ is a primitive $2^n$-th root of unity must be a cyclotomic field. For instance it contains the subfield $\Bbb Q(\cos(2\pi i/2^n))$ which is contained in $\Bbb R$.
By the Kronecker-Weber theorem, the subfields of cyclotomic fields are precisely the finite extensions of $\Bbb Q$ whose Galois group is Abelian. In particular, all quadratic fields $\Bbb Q(\sqrt m)$ for $m\in\Bbb Z$ are contained in cyclotomic fields.