PlanetMath: ideal decomposition in Dedekind domain

(more info)

Math for the people, by the people.

Main Menu

ideal decomposition in Dedekind domain

(Topic)

According to the entry “fractional ideal”, we can state that in a Dedekind domain , each non-zero integral ideal $\mathfrak{a}$ may be written as a product of finitely many prime ideals $\mathfrak{p}_i$ of ,

$\displaystyle \mathfrak{a} = \mathfrak{p}_1\mathfrak{p}_2...\mathfrak{p}_k.$

The product decomposition is unique up to the order of the factors.

Corollary. If $\alpha_1$ , $\alpha_2$ , ..., $\alpha_m$ are elements of a Dedekind domain and is a positive integer, then one has

$\displaystyle (\alpha_1,\,\alpha_2,\,...,\,\alpha_m)^n = (\alpha_1^n,\,\alpha_2^n,\,...,\,\alpha_m^n)$

(1)

for the ideals of

This corollary may be proven by induction on the number of the generators (not on the exponent ).

"ideal decomposition in Dedekind domain" is owned by pahio.

(view preamble)

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

Cross-references: number, induction, ideals, integer, positive, prime ideals, product, integral ideal, Dedekind domain
There is 1 reference to this entry.

This is version 8 of ideal decomposition in Dedekind domain, born on 2005-07-12, modified 2008-04-21.
Object id is 7219, canonical name is IdealDecompositionInDedekindDomain.
Accessed 1176 times total.

Classification:

AMS MSC:	11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
	11R37 (Number theory :: Algebraic number theory: global fields :: Class field theory)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact