Radical and cyclotomic extensions of the rational numbers

Abstract:

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 $|E:K|$ is bounded above by an almost linear function of $|E:R|$. In particular, if $R$ is Galois, it contains a cyclotomic field $K$ such that $|R:K| \le 3$.