Conductor of a ray class field.

Question

I am not getting the definition of Conductor of a ray class field. I know the following definition Let $K$ be a number field. The theorems of class field theory tell us that given any modulus $\mathfrak{m}$ for $K$, there is a unique Abelian extension $K_{\mathfrak{m}}$ such that the kernel of the Artin map of $K_{\mathfrak{m}}/K$ with respect to $\mathfrak{m}$ is precisely the subgroup of principal fractional ideals congruent to $1 \pmod{\mathfrak{m}}$. This is the Ray class field.

1. Could any one give an idea of conductor of ray class field?
2. How the conductor of ray class field are generated?
3. or any material explaining this?

Thank you very much in advance

Answer 1

Ray class field modulo $\mathfrak{m}$ is the maximal abelian extension of conductor $\mathfrak{m}$. The ray class field of conductor $m=1$ is the Hilbert Class field. In particular Hilbert Class field is the ring class field of maximal order in general ring class field of an order of conductor $\mathfrak{m}$ is the intermediate between the Hilbert Class field and the ray class field of conductor $\mathfrak{m}$.

Answer 2

Maybe a bit late but "History of Class Field Theory" was quite enlightening to me. http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/cfthistory.pdf

Nancy Childress, "Class Field Theory" book is very clear and should help you getting throught this "conductor" business more easily.