To a theorem of Isaacs on the casus irreducibilis phenomenon (Q1590709)

Let \(f\) be a cubic polynomial with only real roots. We know that each root is a sum of non-real roots. The author generalizes this observation as follows. Let \(\Omega\) be a field of characteristic \(\neq 0\) and let \(K\) be a subfield which contains for each prime \(p\) the same \(p\)th roots of unity as \(\Omega\) does. Let \(a\) be an algebraic element of \(\Omega/K\) which is contained in a repeated radical extension of \(K\) inside \(\Omega\). If the normal hull \(L\) of \(a\) over \(K\) is contained in \(\Omega\), then for each prime divisor of \([L:K]\) the \(p\)th roots of unity are contained in \(K\).