[GAP Forum] Finding roots of a polynomial

[joyner.david at comcast.net]

Dear GAP Forum:

Frank Luebeck gave one answer to Dursun's question.
Here is an answer to a related, more general question.

 ++++++++++++++++++++++++++++++++++++++++++++++++++
 
 Here's an example
 

 p:=3; n:=4; F:=GF(p^n); c:=Random(F); r:=2;
 x:=X(F,1); f:=x^r-c*x+c-1;
 F_f:=FieldExtension( F, f );
 alpha:=RootOfDefiningPolynomial(F_f);
 
 #alpha is the desired root.

########## Another solution:

 RootOfPolynomial:=function(f,R)
 local F0,Ff,a;
 F0:=CoefficientsRing(R);
 Ff:=FieldExtension(F0,f);
 a:=RootOfDefiningPolynomial(Ff);
 return a;
end;

# example:
p:=3; n:=4; F:=GF(p^n); c:=Random(F); r:=2;
x:=X(F,1); f:=x^r-c*x+c-1;
R:=PolynomialRing( F, [ x ]);
a:=RootOfPolynomial(f,R);
# check:
Value(f,[x],[a]);

# example where F is not finite:
F:=Rationals;
x:=X(F,1); f:=x^2+x+1;
R:=PolynomialRing( F, [ x ]);
a:=RootOfPolynomial(f,R);
# check:
Value(f,[x],[a]);



Hope this helps.
 
 +++++++++++++++++++++++++++++++++++++++++++++++++++
 
 Bulutoglu Dursun A Civ AFIT/ENC wrote:
 
>	Dear Gap Forum,
>	I would like to first thank you for your help before.
>	Now I am trying to solve the following problem:
>       Solve an equation of the form x^r-c*x+c-1=0 in GF(p^n) for any non-zero 
constant c.
> 
>       Dursun.


> 	Dear Gap Forum;
> 	I was wondering whether there is a function in GAP that does the
> following:
> 	
> Input: a list of numbers
> 	Output: the numbers in the list sorted from small to large and
> the frequencies at which they appear in the list.
> 	Dursun.
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