Quadratic subfield of a prime cyclotomic field
= Quadratic subfield of a prime cyclotomic field
{title2=$\mathbb Q(\sqrt{(-1)^{(\ell-1)/2}\ell})$}
For an odd prime $\ell$, the unique quadratic subfield of $\mathbb Q(\zeta_\ell)$ is the displayed field. A quadratic Gauss sum squares to $(-1)^{(\ell-1)/2}\ell$, and the cyclic <Galois group> has a unique subgroup of index two.