Dear GAP-Forum,
Alexandra Alecu asked:
Anyway... my problem is (and it might be an easy one) how to find the
primitive n-th root of unity in a finite field?
I noticed that for the complex numbers I could use E(n), I need
something
similar for a finite field.
All I have at the moment is a way to find the primitive n-th root
of unity
for the cases when n = p^m - 1 where p is a prime and m > 1. For
this I use
the PrimitiveRoot(GF(p^m)) which returns the primitive root of the
finite
field GF(p^m).
The easiest way to find such a root is as power of a primitive root
in a suitable field (which you also need to find):
Given your prime p and n, find the smallest m such that n divides p^m-1:
m:=1;
while (p^m-1) mod n<>0 do
m:=m+1;
od;
Then take the generator of the corresponding finite field (which has
order p^m-1) and power it up to get a primitive root of order n:
a:=PrimitiveRoot(GF(p^m))^((p^m-1)/n);
Be aware that there are some limits on finite fields in GAP, e.g. you
might not be able to do this for arbitrary large p or n.
Best wishes,
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: [EMAIL PROTECTED], Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke
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