Without going into detail, this is due to the fact that the polynomials have another presentation
over Galoisfields and different algorithms / implementations are involved. Also factorize is not at
your disposal.
Note that in general
f/g only gives the quotient without remainder. Most likely that is not what
you want, but you are not lost here.
There is the command
division http://www.singular.uni-kl.de/Manual/latest/sing_226.htm#SEC266which performs the task. It is important to use a global odering
dp as you do.
Code:
factorize(f);
? not implemented
? error occurred in or before STDIN line 6: `factorize(f);`
> list L = division(f,g);
> L;
[1]:
_[1,1]=x-1
[2]:
_[1]=a16
[3]:
_[1,1]=1
> typeof(L[1]);
matrix
> typeof(L[1][1,1]);
poly
> typeof(L[2]);
ideal
> f==g*L[1][1,1] + L[2][1];
1
> (x-1)*(x+1) + 2;
x2+1
> a16;
a16
> number(2);
a16
(The third value
L[3] is a unit matrix in global ordering, but the result is different
if you would work in
ring rads = (49a,),x,ds;Try it!)
You may also define this finite field as an quadratic extension of Z_7 by the
minimal polynomial displayed from the ring itself.
Code:
> setring r;
> basering;
// coefficients: ZZ/49[a]
// minpoly : 1*a^2+6*a^1+3*a^0
// number of vars : 1
// block 1 : ordering dp
// : names x
// block 2 : ordering C
With this approach, the elements are not represented as a power of the primitive
element
a, but now f/g and factorize work.
Code:
> ring ra49 = (7,a),x,dp; minpoly = a2+6a+3;
// compare with above
> a16;
2
> number(2);
2
> poly f = x2+1;
> poly g = x+1;
> f/g;
x-1
> factorize (f);
[1]:
_[1]=1
_[2]=x+(-a-3)
_[3]=x+(a+3)
[2]:
1,1,1
> division(f,g);
[1]:
_[1,1]=x-1
[2]:
_[1]=2
[3]:
_[1,1]=1
> f == (x-1)*g +2;
1
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