Bill Allombert on Sat, 25 Jun 2016 14:21:16 +0200

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Re: p-adic logarithm

On Thu, Jun 23, 2016 at 06:49:07PM +0200, Bill Allombert wrote:
> On Thu, Jun 23, 2016 at 02:48:00PM +0000, LECOUTURIER Emmanuel wrote:
> > Hello,
> > Is there any function which allows us to compute directly Iwasawa
> > p-adic logarithm in finite extensions of Q_p ? (like Q_p(zeta_p))
> GP itself only handles Q_p, however you can factor x as
> x = t*(1+u) where t is of torsion and u is in the convergence domain of
> log(1+X) and then use the power series for log (or for atanh).

Writing this gives me an idea: instead of computing t, if n is the order of
the torsion subgroup, then x^n = (1+u)^n = 1+v and 
log(x^n)= log(x)/n
In some case this can be simpler than actually computing t.