How do I calculate $g_{(X)}\in F_{q}[x]$ if we have $f_{(x)} = g_{(x)}^p$ and $p$ is the characteristic of the field $F$?
This problem arises from the factorization of a polynomial into irreducible polynomials. In case we have $f_{(x)}={g_{(x)}^ p}$, then $$GCD(f_{(x)},f'_{(x)})=f_{(x)}$$
because $f'_{(x)}=0. $ In this case, how can we have the polynomial whose irreducible polynomials is of degree 1?
I figured out myself.
In every $F[x]$ we have $f_{(x)}^p=f_{(x^p)}$ where p is the characteristic of the field. Hence, $$g_{(x)}=f_{(x^{1/p})}$$
Now we can compute that easily.