= Quadratic Kronecker–Weber theorem by ramification
{title2=$\mathbb Q(\sqrt D)\subseteq\mathbb Q(\zeta_{|D|})$}
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 $\mathbb Q(\sqrt{\ell^*})$, where $\ell^*=(-1)^{(\ell-1)/2}\ell$. A fundamental <discriminant> factors as $D=D_2\prod_{\ell\mid D,\ \ell\text{ odd}}\ell^*$ with $D_2\in\{1,-4,8,-8\}$. The remaining <quadratic number field> is generated by $i$, $\sqrt2$, or $\sqrt{-2}$ and lies in $\mathbb Q(\zeta_8)$. Taking the corresponding <compositum> proves the indicated quadratic case of the <Kronecker–Weber theorem>.
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