[GAP Forum] An Equation over a finite field involving one of the primitive elements.

[Frank Luebeck]

On Sat, 1 Jan 2005, Bulutoglu Dursun A Civ AFIT/ENC wrote:

>> x^r-A((p^n-1)/4)x+A((p^n-1)/4)-1=0  where 0=<r=<p^n-1 and
>> A is a non-zero constant in GF(p^n).
>> 
>> 
> Is c=A((p^n-1)/4) a constant, so you are trying to solve x^r-c*x+c-1=0
> in GF(p^n)?
> 
> That is right. I should have stated the problem like that as A can be
> any constant. Sorry about that.
> So now the problem is: If x is a primitive element in GF(p^n) it can not
> solve an equation of the form x^r-c*x+c-1=0 in GF(p^n) for any non-zero
> constant c.

Dear Dursun,

This statement is not true. Try the following in GAP:
     x := Z(3^2);
     Order(x);
     c := Z(3^2)^3;
     x^7 - c*x + c-1;

You could use GAP for checking such statements in some examples. Here
the function

     findexample := function(p, n)
       local y, primelts, z, i, r, c, pol, test;
       y := Indeterminate(GF(p^n), 1);
       primelts := Filtered(GF(p^n), x-> Order(x)=p^n-1);;
       for r in [0..p^n-1] do
         z := Z(p^n);
         for i in [0..p^n-2] do
           c := z^i;
           pol := y^r - c*y +(c-1);
           test := Filtered(primelts, a-> IsZero(Value(pol, a)));
           if Length(test) > 0 then
             Print("r=",r,"  c=",c,"  #solutions=",Length(test),
                   "(",test,")\n");
           fi;
         od;
       od;
     end;

gives you plenty of further counterexamples to your statement.

Best regards,

     Frank

///  Dr. Frank Lübeck, Lehrstuhl D für Mathematik, Templergraben 64,  ///
\\\                    52062 Aachen, Germany                          \\\
///  E-mail: Frank.Luebeck at Math.RWTH-Aachen.De                        ///
\\\  WWW:    http://www.math.rwth-aachen.de/~Frank.Luebeck/           \\\
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