Radical and cyclotomic extensions of the rational numbers

A radical extension of the rational numbers $\mathbb{Q}$ is a field $R \supseteq \mathbb{Q}$ generated by an element having a power in $\mathbb{Q}$, and a cyclotomic extension $K \supseteq \mathbb{Q}$ is an extension generated by a root of unity. We show that a radical extension that is almost Galois over $\mathbb{Q}$ is almost cyclotomic. More precisely, we prove that if $R$ is radical with Galois closure $E$, then $E$ contains a cyclotomic field $K$ such that the degree $\vert E:K\vert$ is bounded above by an almost linear function of $\vert E:R\vert$. In particular, if $R$ is Galois, it contains a cyclotomic field $K$ such that $\vert R:K\vert \le 3$.