Representation of -1 in Galoisfields

[gorzel]

In Galoisfields, all nonzero elements, different from 1 are represented as powers of the genertor.

Are they?

Let N = pⁿ

and xi be a generator of the units GF(N)^x.

If p = 2, then -1 == 1.

If p> 2, then -1 = xi^(N-1)/2.

Singular, however, does not represent -1 as a power, if p > 2.

> ring r25 = (25,a),x,dp;
> a^11;
a11
> a^12;     // (a^12)^2 == a^24 == 1 => a^12 ==-1 
-1
> a^13;
a13

> x+3;
x+a18
> x+4;       // Shouldn't this be displayed as
x-1          // x+a12   ?!
> x+6;
x+1
> x+7;
x+a6

[seelisch]

we change the online docu (if any concerning -1 in Galois fields)