Remark

Let $a \in K^*$ and let r be the least integer $> 0$ such that $a' \in K^n$. Let B be the set of roots in K of the polynomial $X^{n/r} - a$; then we have

by substitution of $X'$ for T in the relation $T^{n/r} - a = \Pi (T - b)$. By Example 4 each of the polynomials $X' - b$ is irreducible, so that (14) is the decomposition of $X^n - a$ into irreducible polynomials in $K[X]$.