= Subfields of the fifteenth cyclotomic field
{title2=$\operatorname{Gal}(\mathbb Q(\zeta_{15})/\mathbb Q)\simeq C_2\times C_4$}
The <Cyclotomic character of a finite cyclotomic extension> and the <Chinese remainder theorem> give the displayed <Galois group>. It has three order-two <subgroups> and three order-four <subgroups>; the latter are kernels of its three nonzero <group homomorphisms> to $C_2$. By the <Fundamental theorem of Galois theory>, its proper intermediate <fields> strictly above $\mathbb Q$ are the three quadratic <fields>
$$
\mathbb Q(\sqrt{-3}),\quad\mathbb Q(\sqrt5),\quad\mathbb Q(\sqrt{-15})
$$
and the three quartic <fields>
$$
\mathbb Q(\zeta_{15}^3),\quad\mathbb Q(\sqrt{-3},\sqrt5),\quad\mathbb Q(\zeta_{15}+\zeta_{15}^{-1}).
$$
The first quartic has cyclic <Galois group> of order four and unique quadratic <subfield> $\mathbb Q(\sqrt5)$; the second is a <biquadratic extension>, so they differ. The last is real whereas the others are not, by the <maximal real subfield of a cyclotomic field> formula. Also $\sqrt{-3}+\sqrt5$ generates the second quartic: if $s=\sqrt{-3}+\sqrt5$, then $\sqrt{-3}=(s-8/s)/2$ and $\sqrt5=(s+8/s)/2$.
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