modulus
A modulus^{} for a number field^{} $K$ is a formal product
$$\underset{\mathrm{\pi \x9d\x94\xad}}{\beta \x88\x8f}{\mathrm{\pi \x9d\x94\xad}}^{{n}_{\mathrm{\pi \x9d\x94\xad}}}$$ 
where

β’
The product is taken over all finite primes and infinite primes of $K$

β’
The exponents ${n}_{\mathrm{\pi \x9d\x94\xad}}$ are nonnegative integers

β’
All but finitely many of the ${n}_{\mathrm{\pi \x9d\x94\xad}}$ are zero

β’
For every real prime $\mathrm{\pi \x9d\x94\xad}$, the exponent ${n}_{\mathrm{\pi \x9d\x94\xad}}$ is either 0 or 1

β’
For every complex prime $\mathrm{\pi \x9d\x94\xad}$, the exponent ${n}_{\mathrm{\pi \x9d\x94\xad}}$ is 0
A modulus can be written as a product of its finite part
$$\underset{\mathrm{\pi \x9d\x94\xad}\beta \x81\u2019\text{\Beta finite}}{\beta \x88\x8f}{\mathrm{\pi \x9d\x94\xad}}^{{n}_{\mathrm{\pi \x9d\x94\xad}}}$$ 
and its infinite part
$$\underset{\mathrm{\pi \x9d\x94\xad}\beta \x81\u2019\text{\Beta real}}{\beta \x88\x8f}{\mathrm{\pi \x9d\x94\xad}}^{{n}_{\mathrm{\pi \x9d\x94\xad}}},$$ 
with the finite part equal to some ideal in the ring of integers^{} ${\mathrm{\pi \x9d\x92\u037a}}_{K}$ of $K$, and the infinite part equal to the product of some subcollection of the real primes of $K$.
Title  modulus 

Canonical name  Modulus 
Date of creation  20130322 12:35:26 
Last modified on  20130322 12:35:26 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  4 
Author  djao (24) 
Entry type  Definition 
Classification  msc 11R37 