Exercise 5, § 11

Let K be a field of characteristic p and n an integer not a multiple of p. For a polynomial of $K[X]$ of the form $X^n - a$ to be irreducible it is necessary and sufficient that for every prime factor q of n, a should not equal the q-th power of an element of K and further when $n \equiv 0 \pmod{4}$, a should not be of the form $-4c^4$ with $c \in K$. (To prove the sufficiency of the condition, reduce to the case where $n = q^e$ with q prime and use Ex. 4 of V, p. 153. Argue by induction an e, determining, with the help of Ex. 4 of V, p. 153, the form of the constant term of each irreducible factor of $X^{q^e} - a$.)