ideal decomposition in Dedekind domain
According to the entry “fractional ideal (http://planetmath.org/FractionalIdeal)”, we can that in a Dedekind domain , each non-zero integral ideal may be written as a product of finitely many prime ideals of ,
The product decomposition is unique up to the order of the factors. This is stated and proved, with more general assumptions, in the entry “prime ideal factorisation is unique (http://planetmath.org/PrimeIdealFactorizationIsUnique)”.
Corollary. If , , …, are elements of a Dedekind domain and is a positive integer, then one has
for the ideals of .
This corollary may be proven by induction on the number of the ).
|Title||ideal decomposition in Dedekind domain|
|Date of creation||2015-05-05 19:05:43|
|Last modified on||2015-05-05 19:05:43|
|Last modified by||pahio (2872)|