Quadratic Kronecker–Weber theorem by ramification ($\mathbb{Q}(\sqrt{D})\subseteq \mathbb{Q}({\zeta }_{|D|})$)

For an odd prime $\ell$, the unique quadratic subfield of a prime cyclotomic field has no finite ramification outside $\ell$. The classification of quadratic discriminants identifies it with $\mathbb{Q}(\sqrt{{\ell }^{\ast }})$, where ${\ell }^{\ast }=(-1{)}^{(\ell -1)/2}\ell$. A fundamental discriminant factors as $D=D_2{\prod }_{\ell \mid D,\ell \text{ odd}}{\ell }^{\ast }$ with $D_2\in \{1,-4,8,-8\}$. The remaining quadratic number field is generated by $i$, $\sqrt{2}$, or $\sqrt{-2}$ and lies in $\mathbb{Q}({\zeta }_8)$. Taking the corresponding compositum proves the indicated quadratic case of the Kronecker–Weber theorem.