Singular

Thank you for your answer, this definitely solves my problem :)

But is this kind of pitfall anywhere documented? I tried to find something, but using google to find problems with Singular is kind of hard.

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 [i]f/g[/i] only gives the quotient without remainder. Most likely that is not what
you want, but you are not lost here.
There is the command [b]division[/b]
[url]http://www.singular.uni-kl.de/Manual/latest/sing_226.htm#SEC266[/url]
which performs the task. It is important to use a global odering [b]dp[/b] 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
[/code]
(The third value [i]L[3][/i] is a unit matrix in global ordering, but the result is different
if you would work in [b]ring rads = (49a,),x,ds;[/b]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
[/code]
With this approach, the elements are not represented as a power of the primitive
element [i]a[/i], 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
[/code]

Why is it not possible to divide polynomials over GF(p^n), if n >= 2? I get the following error:
[code]> ring r = (7^2,a),x,dp;
> poly f = x2+1;
> poly g = x+1;
> f/g;
? not implemented
? error occurred in or before STDIN line 14: `f/g;`
[/code]

Any help would be appreciated.