Re: SNF over a PID

[Alice Mark <amark@math.utexas.edu>]

Thanks for your response on this.  I have a follow up question.  Is 
there a way to have the output be similar to matsnf with the flag 1?

Alice

On 11/29/2012 02:28 PM, Bill Allombert wrote:
> On Thu, Nov 29, 2012 at 01:13:29PM -0600, Alice Mark wrote:
> > Hello pari-users,
> >
> > I would like to know if there is a way to find the Smith Normal Form
> > of a matrix over a PID other than the integers.  matsnf() seems to
> > only work on integer matrices.  Specifically, I'd like to do this in
> > an integer ring of some quadratic extension of Q (in a case where
> > that's a PID).
> Hello Alice,
>
> Note that matsnf also work for polynomial rings over field (which are PID).
>
> In your case, you need to use nfsnf which also handles non principal
> integer rings, hence has some complication, which you can mostly ignore.
> (in particular, you can take all the b_i and c_i to be 1).
>
> Here an example (where x denotes I)
>
> ? B=bnfinit(x^2+1);
> ? nfsnf(B,[[1,x;1,1],[1,1],[1,1]])
> %6 = [[2,1;0,1],[1,0;0,1]]
> ? apply(x->bnfisprincipal(B,x)[2],%)
> %7 = [[1,1]~,[1,0]~]
>
> Since the integral basis is [1,I], the HNF is [1+I,1]
>
> Cheers,
> Bill.