Fractional ideal

A fractional ideal of a number field is a nonzero

O_{K}

I \subset K

for which

d I \subseteq O_{K}

for some nonzero

d \in O_{K}

.

Table of contents
- Principal fractional ideal Fractional ideal
- Integral ideal Fractional ideal

Principal fractional ideal ( $a O_{K}$ )

A principal fractional ideal of a number field is

a O_{K}

for a nonzero

a \in K

. Its generators differ by a unit in the ring of integers of a number field. Quotienting nonzero fractional ideals by principal fractional ideals gives the ideal class group; requiring congruences and signs of the generators instead gives a ray class group.

Integral ideal

An integral ideal is a fractional ideal contained in the ring of integers

O_{K}

.

 Ancestors (5)

 Incoming links (25)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Fractional ideal by

Wikipedia Bot  1

In the context of algebraic number theory, a **fractional ideal** is a generalization of the notion of an ideal in a ring. Specifically, fractional ideals are particularly useful in the study of Dedekind domains and more generally in the structure of arithmetic in number fields. ### Definitions and Properties 1. **Integral Domain**: First, consider a domain \( R \), typically a Dedekind domain, which is an integral domain where every nonzero proper prime ideal is maximal.

 Read the full article

  See all articles in the same topic Create my own version