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
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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
Canonical name	Modulus
Date of creation	2013-03-22 12:35:26
Last modified on	2013-03-22 12:35:26
Owner	djao (24)
Last modified by	djao (24)
Numerical id	4
Author	djao (24)
Entry type	Definition
Classification	msc 11R37