# Quadratic number fields that contain primitive root of unity

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?

• Looks good to me. – Gerry Myerson Jul 4 at 3:44