Solvable primitive extensions

Abstract

A finite separable extension $E$ of a field $F$ is called primitive if there are no intermediate extensions. It is called solvable if the group $\mathrm{Gal}(\hat E|F)$ of automorphisms of its galoisian closure $\hat E$ over $F$ is solvable, and a $p$-extension ($p$ prime) if the degree $[E:F]$ is a power of $p$. We show that a solvable primitive $p$-extension $E$ of $F$ is uniquely determined (up to $F$-isomorphism) by $\hat E$ and characterise the extensions $D$ of $F$ such that $D=\hat E$ for some solvable primitive $p$-extension $E$ of $F$.