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 $F$.

6 pages

References in corpus (4)

• Evolution of proto-galaxy-clusters to their present form: theory and observations
• Little galoisian modules
• Wildly primitive extensions
• ${\mathbf{F}}_p$-representations over $p$-fields

Cited by in corpus (2)

• Wildly primitive extensions
• ${\mathbf{F}}_p$-representations over $p$-fields