Let $f$ be arbitrary polynomial over a finite field.The Galois group over the finite field is ,as is well known,cyclic,and the generator is the problem. I heard $f$'s Galois group is completely determined, or a generator is determined, when $f$'s irreducible factorization is given.
But I cannot remember the correct statement and its proof, may be the generator is some product of cyclic permutations. Could you give me the correct statement and proof regarding this?
Reference（webpages, books,etc）is also appreciated. Thank you for your kind help.