isPrimitive -- whether an element is a primitive element of a finite field

isPrimitive -- whether an element is a primitive element of a finite field

isPrimitive(f) -- Given an element f in a quotient of a polynomial ring R over a finite field Kwhich is itself a finite field, with the ring being finite dimensional over the field, determine if f generates the multiplicative group of this field.

i1 : R = ZZ/5[t]/(t^2+t+1);

i2 : isPrimitive t



o2 = false

i3 : isPrimitive (t-1)



o3 = true

Code:

     -- ../../../Macaulay2/m2/galois.m2:47-53
     isPrimitive = (g) -> (
          g != 0 and
          unpack(ring g,
               (R,p,n,f) -> (
                    q := p^n;
                    all(factor (q-1), v -> 1 != g ^ ((q-1)//v#0))
                    )))