
Theorem
I have found this thoerem in a book but its proof is not clear , Please can any one explain it to me step by step or give me an easy source that has it.
Theorem:
let t be a positive integer , and P(x) belongs to F_q [x] be irreducible over F_q of degree n , not of the form cx ,and period e (which equals the order of any root of P(x))
Then P(x^t) is irreducible over F_q if and only if
(i) Each prime divisor of t divides e,
(ii) gcd (t, (q^n 1 )/e) =1 , and
(iii) if 4t then 4 (q^n 1).
Thanks