ray class field

Proposition 1.

Let $L\mathrm{/}K$ be a finite abelian extension of number fields, and let${\mathrm{O}}_K$ be the ring of integers of $K$. There exists anintegral ideal $\mathrm{C}\mathrm{\subset }{\mathrm{O}}_K$, divisible byprecisely the prime ideals of $K$ that ramify in $L$, such that

$$((\alpha ),L/K)=1,\forall \alpha \in K^{\ast },\alpha \equiv 1\mathrm{mod}𝒞$$

where$\mathrm{(}\mathrm{(}\alpha \mathrm{)}\mathrm{,}L\mathrm{/}K\mathrm{)}$ is the Artin map.

Definition 1.

The conductor of a finite abelian extension $L\mathrm{/}K$ is thelargest ideal ${\mathrm{C}}_{L\mathrm{/}K}\mathrm{\subset }{\mathrm{O}}_K$ satisfyingthe above properties.

Note that there is a “largest ideal” with this condition becauseif proposition 1 is true for ${𝒞}_1,{𝒞}_2$ then itis also true for ${𝒞}_1+{𝒞}_2$.

Definition 2.

Let $\mathrm{I}$ be an integral ideal of $K$. A ray classfield of $K$ (modulo $\mathrm{I}$) is a finite abelian extension$K_{\mathrm{I}}\mathrm{/}K$ with the property that for any other finiteabelian extension $L\mathrm{/}K$ with conductor ${\mathrm{C}}_{L\mathrm{/}K}$,

$${𝒞}_{L/K}\mid ℐ\Rightarrow L\subset K_{ℐ}$$

Note: It can be proved that there is a unique ray class field witha given conductor. In words, the ray class field is the biggestabelian extension of $K$ with a given conductor (although theconductor of $K_{ℐ}$ does not necessarily equal$ℐ$ !, see example $2$).

Remark: Let $𝔭$ be a prime of $K$ unramified in $L$, and let $𝔓$ be a prime above $𝔭$. Then $(𝔭,L/K)=1$ if and only if the extension of residue fields is of degree 1

$$[{𝒪}_L/𝔓:{𝒪}_K/𝔭]=1$$

if and only if $𝔭$ splits completely in $L$. Thus we obtain a characterization of the ray class field of conductor $𝒞$ as the abelian extension of $K$ such that a prime of $K$ splits completely if and only if it is of the form

$$(\alpha ),\alpha \in K^{\ast },\alpha \equiv 1\mathrm{mod}𝒞$$

Examples:

1. The ray class field of $\mathbb{Q}$ of conductor $N\mathbb{Z}$ is the$N^{th}$-cyclotomic extension of $\mathbb{Q}$. More concretely, let${\zeta }_N$ be a primitive $N^{th}$ root of unity. Then

   $${\mathbb{Q}}_{N\mathbb{Z}}=\mathbb{Q}({\zeta }_N)$$

2. $$\mathbb{Q}{(i)}_{(2)}=\mathbb{Q}(i)$$

   so the conductor of$\mathbb{Q}{(i)}_{(2)}/\mathbb{Q}$ is $(1)$.

3. $K_{(1)}$, the ray class field of conductor $(1)$, is themaximal abelian extension of $K$ which is unramified everywhere.It is, in fact, the Hilbert class field of $K$.