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 an integral ideal $\mathrm{C}\mathrm{\subset}{\mathrm{O}}_{K}$, divisible by precisely the prime ideals^{} of $K$ that ramify in $L$, such that
$$((\alpha ),L/K)=1,\forall \alpha \in {K}^{\ast},\alpha \equiv 1\mathrm{mod}\mathcal{C}$$ 
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 the largest ideal ${\mathrm{C}}_{L\mathrm{/}K}\mathrm{\subset}{\mathrm{O}}_{K}$ 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 $\mathrm{I}$ be an integral ideal of $K$. A ray class field of $K$ (modulo $\mathrm{I}$) is a finite abelian extension ${K}_{\mathrm{I}}\mathrm{/}K$ with the property that for any other finite abelian extension $L\mathrm{/}K$ with conductor ${\mathrm{C}}_{L\mathrm{/}K}$,
$${\mathcal{C}}_{L/K}\mid \mathcal{I}\Rightarrow L\subset {K}_{\mathcal{I}}$$ 
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 $\U0001d52d$ be a prime of $K$ unramified in $L$, and let $\U0001d513$ be a prime above $\U0001d52d$. Then $(\U0001d52d,L/K)=1$ if and only if the extension^{} of residue fields is of degree 1
$$[{\mathcal{O}}_{L}/\U0001d513:{\mathcal{O}}_{K}/\U0001d52d]=1$$ 
if and only if $\U0001d52d$ 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
$$(\alpha ),\alpha \in {K}^{\ast},\alpha \equiv 1\mathrm{mod}\mathcal{C}$$ 
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 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  20130322 13:54:01 
Last modified on  20130322 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 