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

[Bulutoglu Dursun A Civ AFIT/ENC]

>	Dear Gap Forum,
>	I would like to first thank you for your help before.
>	Now I am trying to solve the following problem:
>	
>	Let x be a primitive element of GF(p^n)  for n>=0.
>	Then x can not satisfy an equation of the form:
>	
>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.

Dursun.


>Observe that r=0 or r=p^n-1 implies x=1. I would greatly appreciate it
>if someone could give me an insight how this problem could be solved.	
>Dursun.
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