modulus

A modulus for a number field $K$ is a formal product

\\prod_{\\mathfrak{p}}\\mathfrak{p}^{n_{\\mathfrak{p}}}

where

•
The product is taken over all finite primes and infinite primes of $K$
•
The exponents $n_{\\mathfrak{p}}$ are nonnegative integers
•
All but finitely many of the $n_{\\mathfrak{p}}$ are zero
•
For every real prime $\\mathfrak{p}$ , the exponent $n_{\\mathfrak{p}}$ is either 0 or 1
•
For every complex prime $\\mathfrak{p}$ , the exponent $n_{\\mathfrak{p}}$ is 0

A modulus can be written as a product of its finite part

\\prod_{\\mathfrak{p}\\text{ finite}}\\mathfrak{p}^{n_{\\mathfrak{p}}}

and its infinite part

\\prod_{\\mathfrak{p}\\text{ real}}\\mathfrak{p}^{n_{\\mathfrak{p}}},

with the finite part equal to some ideal in the ring of integers $\\mathcal{O}_{K}$ of $K$ , and the infinite part equal to the product of some subcollection of the real primes of $K$ .