Hey PF! Alright so the Galois correspondence links intermediate fields to subgroups of the Galois group. Let's call * the map that takes an intermediate field to it's subgroup and " the map that takes a Galois subgroup to it's intermediate field. I need to find an example for a subgroup H such that H"* does not equal H. H will always be included in H"* but will only equal H under some special circumstances that I don't completely understand, I don't think it's as easy as if the field extension is Galois.
The Attempt at a Solution
I've tried quite a few things. I started off with something basic like Q(a,c):Q where a is a 3rd cube root of 2 and c is a primitive fourth root or something like that. I figure my best luck will be if I look for extensions that aren't galois. Then I tried intermediate fields Q(ac), Q(a), Q(c) but couldn't find anything who's subgroup it was mapped to had the property I was looking for, which is again H"* not equalling H. If anyone has any tips for me that'd be awesome.