Subfields of a cyclotomic field 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.

Let $f(x)$ be a polynomial with integer coefficients. For a root of unity $\zeta$, the subfields $\mathbf{Q}[f(\zeta +\bar\zeta )]\subset \mathbf{Q}[\zeta]$ give for most polynomials $f$ give examples non-cyclotomic fields.(of course two different choices of $f$ may lead to the same subfield). These are all reals.
When $\zeta$ is a $p$-th root of unity, for a prime number $p$, using Quadratic Gauss sums  Gauss showed (generalizing your example $p=5$) that when $p-1$ is a multiple of $4$, the real quadratic field $\mathbf{Q}[\sqrt p]$ is a subfield of the $p$th cyclotomic field.
