Bill Allombert on Thu, 18 Dec 2014 15:42:38 +0100

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Re: Factoring in unique factorization domains

On Thu, Dec 18, 2014 at 02:50:38PM +0200, Andrew Lelechenko wrote:
> What is the best way to factor integers in quadratic rings, which
> are unique factorization domains, using PARI/GP? Should I use
> idealfactor() with nfinit()?

More precisely, you need to compute bnfinit() so you can use
bnfisprincipal() to compute the generators of the prime ideals.

Note that factor() already handles Z[i] so this can be used as a test.

More generally, you can do the same in non-principal domains to
obtain all the factorization of of number as product of irreducible,
but you need to do some combinatoric in the class group to identify
principal ideals.