# modulus

A 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 $K$.

Title modulus Modulus 2013-03-22 12:35:26 2013-03-22 12:35:26 djao (24) djao (24) 4 djao (24) Definition msc 11R37