ray class field

Proposition 1.

Let $L/K$ be a finite abelian extension of number fields, and let $\mathcal{O}_{K}$ be the ring of integers of $K$. There exists an integral ideal $\mathcal{C}\subset\mathcal{O}_{K}$, divisible by precisely the prime ideals of $K$ that ramify in $L$, such that

 $\left((\alpha),L/K\right)=1,\quad\forall\alpha\in K^{\ast},\ \alpha\equiv 1\ % \operatorname{mod}\ \mathcal{C}$

where $\left((\alpha),L/K\right)$ is the Artin map.

Definition 1.

The conductor of a finite abelian extension $L/K$ is the largest ideal $\mathcal{C}_{L/K}\subset\mathcal{O}_{K}$ satisfying the above properties.

Note that there is a “largest ideal” with this condition because if proposition 1 is true for $\mathcal{C}_{1},\mathcal{C}_{2}$ then it is also true for $\mathcal{C}_{1}+\mathcal{C}_{2}$.

Definition 2.

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

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

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

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

 $[\mathcal{O}_{L}/\mathfrak{P}\colon\mathcal{O}_{K}/\mathfrak{p}]=1$

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

 $(\alpha),\quad\alpha\in K^{\ast},\ \alpha\equiv 1\ \operatorname{mod}\ % \mathcal{C}$

Examples:

1. 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. 2.
 $\mathbb{Q}(i)_{(2)}=\mathbb{Q}(i)$

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

3. 3.

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

References

 Title ray class field Canonical name RayClassField Date of creation 2013-03-22 13:54:01 Last modified on 2013-03-22 13:54:01 Owner alozano (2414) Last modified by alozano (2414) Numerical id 5 Author alozano (2414) Entry type Definition Classification msc 11R37 Synonym conductor Related topic ArtinMap Related topic ExistenceOfHilbertClassField Related topic NumberField Related topic AnExactSequenceForRayClassGroups Defines conductor of an extension