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:

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

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

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

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

3.

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

References

1 Artin/Tate, Class Field Theory. W.A.Benjamin Inc., New York.

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