View source for Modulus in algebraic number theory

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A formal product of [[Place of a field|place]]s of an  [[algebraic number field]], also termed an ''extended ideal''.  It is used to encode [[ramification]] data  for [[abelian extension]]s of a number field (cf [[Conductor of an Abelian extension]]).

Let $K$ be an algebraic number field with ring of integers $R$.  A ''modulus'' is a formal product
$$
\mathfrak{m} = \prod_{\mathfrak{p}} \mathfrak{p}^{\nu(\mathfrak{p})}
$$
where  $\mathfrak{p}$ runs over all places of $K$, finite or infinite, the exponents $\nu$ are zero except for finitely many $\mathfrak{p}$, for  real places $\mathfrak{r}$ we have $\nu(\mathfrak{r})=0$ or $1$ and for  complex places $\nu=0$.

We extend the notion of  [[congruence]] to this setting.  Let $x$ and $y$ be elements of ''K''.   For a finite place $\mathfrak{p}$, that is, a prime ideal of the ring of  integers, we define $x$ and $y$ to be congruent modulo $\mathfrak{p}^n$  if ''x''/''y'' is in the [[valuation ring]] $R_{\mathfrak{p}}$ of  ${\mathfrak{p}}$ and congruent to 1 modulo $\mathfrak{p}^n$ in  $R_{\mathfrak{p}}$ in the usual sense of ring theory.  For a real place  $\mathfrak{r}$ we define $x$ and $y$ to be congruent modulo  $\mathfrak{r}$ if $x/y$ is positive in the real embedding of $K$  associated to the place $\mathfrak{r}$.  Finally, we define $x$ and $y$  to be congruent modulo $\mathfrak{m}$ if they are congruent modulo  $\mathfrak{p}^{\nu(\mathfrak{p})}$ whenever $\nu(\mathfrak{p}) > 0$.

==Ray class group==
We  split the modulus $\mathfrak{m}$ into $\mathfrak{m}_\text{fin}$ and  $\mathfrak{m}_\text{inf}$, the product over the finite and infinite  places respectively.  Define
$$
K_{\mathfrak{m}} = \left\lbrace a/b \in K \mid a,b \in R,~ ab ~\mbox{coprime to}~ \mathfrak{m}_\mbox{fin} \right\rbrace \,,
$$
$$
K_{\mathfrak{m},1} = \left\lbrace x \in K_{\mathfrak{m}} \mid x \equiv 1 \pmod \mathfrak{m} \right\rbrace \ .
$$

We call the group $K_{\mathfrak{m},1}$ the ''ray modulo'' $\mathfrak{m}$.

Further  define the subgroup of the ideal group $I^{\mathfrak{m}}$ to be the  subgroup generated by ideals coprime to $\mathfrak{m}_\text{fin}$.  The  ''ray class group'' modulo $\mathfrak{m}$ is the quotient  $I^{\mathfrak{m}} / i(K_{\mathfrak{m},1})$, where $i$ is the map from  $K$ to principal ideals in the ideal group.  A coset of  $i(K_{\mathfrak{m},1})$ is a ''ray class''.

The ray class group is finite.  Its order is the ''ray class number'': this divides the [[Class number (number theory)|class number]] of $K$.  For the trivial modulus $\mathfrak{m} = 1$, the ray class group is just the [[ideal class group]].

Hecke's  original definition of [[Hecke character]]s may be interpreted in terms  of [[Character|character]]s of the ray class group with respect to some  modulus $\mathfrak{m}$.

==References==
*  Harvey Cohn. "A classical invitation to algebraic numbers and class  fields" (Springer-Verlag, 1978) ISBN 0-387-90345-3.  pp.163-187 
*  Harvey Cohn. "Introduction to the construction of class fields".  Cambridge studies in advanced mathematics '''6''' (Cambridge University  Press, 1985) ISBN 0-521-24762-4. 
* Gerald J. Janusz.  "Algebraic  Number Fields".  Pure and Applied Mathematics '''55''' (Academic Press,  1973) ISBN 0-12-380250. pp.107-113.