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An elliptic curve with complex multiplication is called a CM elliptic curve. Distinguish geometric endomorphisms from endomorphisms defined over the base field. In characteristic zero, adjoining the imaginary quadratic field selected by the tangent action makes all geometric endomorphisms defined over the base. Good split primes have ordinary reduction; good inert primes have supersingular reduction.

An abelian variety of dimension $g$ has complex multiplication when its rational endomorphism ring contains a commutative semisimple $\mathbb{Q}$-algebra of dimension $2g$. For an elliptic curve in characteristic zero, this means its geometric endomorphism ring of an elliptic curve is an order in an imaginary quadratic field. The resulting rank-one Tate module representations connect torsion points to class field theory.

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 ${\mathrm{Cl}}_{\mathfrak{m}}(K)=I_K(\mathfrak{m})/P_K(\mathfrak{m})$.

The classification theorem of class field theory asserts the existence of a ray class field $K_{\mathfrak{m}}$ and an Artin map isomorphismIts 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 $H_K/K$ withIt 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.

A quadratic number field is a number field of degree of a field extension two, necessarily of the form $\mathbb{Q}(\sqrt{d})$ for a square-free integer $d\ne 0,1$. Its two field embeddings send $a+b\sqrt{d}$ to $a\pm b\sqrt{d}$. If $d>0$, both images lie in the real numbers, giving a totally real number field; if $d<0$, they form a pair related by complex conjugation, giving an imaginary quadratic field. The ring of integers of a quadratic field depends on $d$ modulo four.^[ref]

For every positive integer $n$, some imaginary quadratic field has an ideal class of order of a group element greater than $n$. Specifically, take a prime number $p\equiv 1(\mathrm{mod}20)$ with $p>5^n$. In $\mathbb{Z}[\sqrt{-p}]$, the Dedekind factorization theorem gives $(5)=P\overline{P}$, with distinct prime ideals $P=(5,\sqrt{-p}-2)$ and $\overline{P}=(5,\sqrt{-p}+2)$ of ideal norm five. If $P^k=(a+b\sqrt{-p})$ for $1\le k\le n$, the field norm equation $5^k=a^2+p{b}^2$ forces $b=0$. Then $k$ is even and $(a)=(5{)}^{k/2}=P^{k/2}{\overline{P}}^{k/2}$, contradicting unique factorization of ideals in a number field. Thus $[P]$ has order of a group element greater than $n$. The existence of suitable $p$ follows from Dirichlet's theorem on arithmetic progressions.