# Field-theoretic description of fixed field of central subgroups?

Given a Galois extension $E/F$ with Galois group $G$, and a subextension $E/K$ with Galois group $H$, is there a "field-theoretic" characterization of when $H$ is central (i.e. $H\leq Z(G)$)? By "field-theoretic", I mean something like the characterization of $H$ being normal in $G$ as $E$ being the splitting field of some polynomial in $F[X]$.

• Well, since $\,H\le Z(G)\implies H$ is normal, then you can tell in one direction that the subextension $\; E/K\;$ is normal... – DonAntonio Apr 30 '13 at 22:01
• To the best of my knowledge there is no such criterion (and I am a research mathematician who uses a lot of Galois theory). Of course this means I would be delighted to hear of one... – Pete L. Clark Apr 30 '13 at 22:53