Example 4

Conversely let $a \in K^*$ and let $r$ be the least integer $> 0$ such that $a' \in K^{*n}$; then the subfield $L$ of $\Omega$ generated by the roots of the polynomial $X^n - a$ is a cyclic extension of $K$ of degree $r$. In particular, $X^n - a$ is irreducible if and only if $r = n$.