= Solution
For a <number field> $K$ and a <modulus of a number field> $\mathfrak m$, let $I_K(\mathfrak m)$ be the <group> of <fractional ideals> prime to its finite part. Let $P_K(\mathfrak m)$ consist of <principal ideals> generated by elements congruent to one modulo its finite part and positive at each real place in its infinite part. The <ray class group> is $\operatorname{Cl}_{\mathfrak m}(K)=I_K(\mathfrak m)/P_K(\mathfrak m)$.
The classification theorem of <class field theory> asserts the existence of a <ray class field> $K_{\mathfrak m}$ and an <Artin map> <isomorphism>
$$
\operatorname{Cl}_{\mathfrak m}(K)\xrightarrow{\sim}\operatorname{Gal}(K_{\mathfrak m}/K).
$$
Its <intermediate fields> correspond contravariantly to <subgroups> of this <ray class group>. Equivalently, for each <congruence subgroup of fractional ideals> $P_K(\mathfrak m)\subseteq H\subseteq I_K(\mathfrak m)$ there is a unique finite <abelian extension> with Artin <group kernel> $H$ and <group> $I_K(\mathfrak m)/H$. Every finite <abelian extension> occurs for some modulus. For a fixed modulus, these are exactly the extensions whose conductor divides that modulus; the same field may be presented using larger moduli, with the appropriate pulled-back <group kernel>.
We quote the conductor facts needed to specialize this classification. The finite conductor exponent is zero exactly at the unramified primes. A real place occurs in the conductor exactly when it becomes complex. Thus conductor one means unramified at every finite prime and split at every real place. With $\mathfrak m=1$ the <ray class group> is the ordinary <ideal class group>, so classification gives an extension $H_K/K$ with
$$
\boxed{\operatorname{Gal}(H_K/K)\cong\operatorname{Cl}(K),\qquad[H_K:K]=h_K.}
$$
It is unramified at all finite primes, and its real places split. Conversely, every finite <abelian extension> with those properties has conductor one and lies in $K_1=H_K$. This proves the existence and maximality of the ordinary <Hilbert class field>. Its degree is finite by finiteness of the <ideal class group>. The real-place convention matters: allowing real places to become complex gives the narrow class field instead; for the <imaginary quadratic field> in part (ii) this distinction disappears.
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