Modulus (algebraic number theory)

In mathematics, in the field of algebraic number theory, a modulus (or an extended ideal) is a formal product of places of an algebraic number field. It is used to encode ramification data for abelian extensions of number field.

Definition
Let K be an algebraic number field with ring of integers R. A modulus is a formal product


 * $$\mathbf{m} = \prod_{\mathbf{p}} \mathbf{p}^{\nu(\mathbf{p})} $$

where p runs over all places of K, finite or infinite, the exponents ν are zero except for finitely many p, for real places r we have ν(r)=0 or 1 and for complex places ν=0.

We extend the notion of congruence to this setting. Let x and y be elements of K. For a finite place p, that is, a prime ideal of the ring of integers, we define x and y to be congruent modulo pn if x/y is in the valuation ring Rp of p and congruent to 1 modulo pn in Rp in the usual sense of ring theory. For a real place r we define x and y to be congruent modulo r if x/y is positive in the real embedding of K associated to r. Finally, we define x and y to be congruent modulo m if they are congruent modulo pν(p) whenever ν(p) &gt; 0.

Ray class group
We split the modulus m into mfin and minf, the product over the finite and infinite places respectively. Define


 * $$ K_{\mathbf{m}} = \left\lbrace a/b \in K \mid a,b \in R,~ ab ~\mbox{coprime to}~ \mathbf{m}_\mbox{fin} \right\rbrace ,$$
 * $$ K_{\mathbf{m},1} = \left\lbrace x \in K_{\mathbf{m}} \mid x \equiv 1 \pmod \mathbf{m} \right\rbrace .$$

We call the group Km,1 the ray modulo m.

Further define the subgroup of the ideal group Im to be the subgroup generated by ideals coprime to mfin. The ray class group modulo m is the quotient Im / i(Km,1), where i is the map from K to principal ideals in the ideal group. A coset of i(Km,1) is a ray class.

Properties

 * When m = 1, the ray class group is just the ideal class group.
 * The ray class group is finite. Its order is the ray class number.
 * The ray class number divides the class number of K.