PlanetMath: modulus

(more info)

Math for the people, by the people.

Main Menu

modulus

(Definition)

A modulus for a number field is a formal product

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

where

The product is taken over all finite primes and infinite primes of
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

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

and its infinite part

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

with the finite part equal to some ideal in the ring of integers $\mathcal{O}_K$ of

, and the infinite part equal to the product of some subcollection of the real primes of

"modulus" is owned by djao.

(view preamble)

Log in to rate this entry.
(view current ratings)

Cross-references: ring of integers, ideal, infinite, finite, complex prime, real prime, integers, exponents, infinite primes, finite primes, product, number field
There are 3 references to this entry.

This is version 1 of modulus, born on 2002-04-16.
Object id is 2841, canonical name is Modulus.
Accessed 4221 times total.

Classification:

11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact