Find all quadratic fields $\mathbb{Q}[\sqrt{d}]$ that contain some $p$-th primitive root of unity, where $p>2$ is a prime.
Now, my reasoning was: if $\mathbb{Q}[\sqrt{d}]$ contains one $p$-th root of unity, then it contains all of them and $p$-th cyclotomic field is a subfield of $\mathbb{Q}[\sqrt{d}]$. From that we have $2=[\mathbb{Q}[\sqrt{d}]:\mathbb{Q}]\geq p-1>1$, so $p-1$ must be $2$ and $p=3$.
For $p>3$ are no such quadratic number fields and for $p=3$ the required field is 3rd cyclotomic field. Is this correct? Or am I missing something?