Sorry it is Kroneker Webber theorem and it a teorem about abelian extension. But it is how you said. The Galois group of a cyclotomic extenesion Q(\xi) with \xi a p-root of unity is the cyclic group of order p-1. As such it has a unique subgroup of index 2, which corresponds to a unique subestension, quadratic over Q.
In the chapter about abelian extension you can aslo see that this extesnsion is generated by
$\alpha:= \sum_{a\in \mathbb{F}_p^{*2}}\xi^a$
Thus the minimal polynomial of $\alpha$ is $(x-\alpha)(x-\overline{\alpha})$ which can be explicitly computed and then one finds that $\alpha\i\mathbb{Q}(\sqrt{p})$ or $\alpha\i\mathbb{Q}(i\sqrt{p})$ depending from the remainder class of $p$ mod 4.