polrootsmod

Function: polrootsmod
Section: polynomials
C-Name: polrootsmod
Prototype: GDG
Help: polrootsmod(f,{D}): roots of the polynomial f over the finite field
 defined by the domain D.
Doc: vector of roots of the polynomial $f$ over the finite field defined
 by the domain $D$ as follows:

 \item $D = p$ a prime: factor over $\F_p$;

 \item $D = [T,p]$ for a prime $p$ and $T(y)$ an irreducible polynomial over
 $\F_p$: factor over $\F_p[y]/(T)$ (as usual the main variable of $T$
 must have lower priority than the main variable of $f$);

 \item $D$ a \typ{FFELT}: factor over the attached field;

 \item $D$ omitted: factor over the field of definition of $f$, which
 must be a finite field.

 \noindent Multiple roots are \emph{not} repeated.
 \bprog
 ? polrootsmod(x^2-1,2)
 %1 = [Mod(1, 2)]~
 ? polrootsmod(x^2+1,3)
 %2 = []~
 ? polrootsmod(x^2+1, [y^2+1,3])
 %3 = [Mod(Mod(1, 3)*y, Mod(1, 3)*y^2 + Mod(1, 3)),
       Mod(Mod(2, 3)*y, Mod(1, 3)*y^2 + Mod(1, 3))]~
 ? polrootsmod(x^2 + Mod(1,3))
 %4 = []~
 ? liftall( polrootsmod(x^2 + Mod(Mod(1,3),y^2+1)) )
 %5 = [y, 2*y]~
 ? t = ffgen(y^2+Mod(1,3)); polrootsmod(x^2 + t^0)
 %6 = [y, 2*y]~
 @eprog