Bill Allombert on Fri, 07 Nov 2014 11:36:39 +0100

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Re: Cube roots mod prime powers

On Fri, Nov 07, 2014 at 12:03:11AM -0500, Charles Greathouse wrote:
> Is there a way to find cube roots mod prime powers, rather than just
> primes? Modular roots are handled by gen_Shanks_sqrtn, which is quite
> specific to primes, but I don't know if there's away to do Hensel lifting
> other than 'by hand'.

Use sqrtn and t_PADIC:
? sqrtn(17+O(5^10),3)
%2 = 3+4*5+2*5^2+4*5^3+4*5^4+4*5^5+5^6+5^7+2*5^8+3*5^9+O(5^10)

> Also, is there a way to get Fp_sqrtn's *zeta from gp? gpow just throws it
> away, but it would be nice to know what the other roots were. (At the
> moment I'm using ./gp-sta because I'm on a temporary system rather than my
> usual setup, so solutions that don't use install() would be best. I'm not
> even sure if you can access that sort of parameter with install, though.)

Again, use sqrtn:
? sqrtn(Mod(2,31),3,&z)
%7 = Mod(4,31)
? z
%8 = Mod(25,31)