Where is wrong in my proof? For all finite dimensional field extension we have $H’’=H$ for all subgroups of Galois group.

Question

$F$ is a finite dimensional field extension of $K$. $G$ is the Galois group of $F$. For all intermediate field $E$, $E’$ is the fixing group and for all subgroup $H$ of $G$, $H’$ is the fixed field. Let $0$ be the trivial subgroup and then $0’ = F, 0’’=F’=0$ Since $H \le H’’$ we have $$\vert H \vert \le \vert H’’ \vert = [H’’ : 0’’] \le [0’:H’] \le [H:0] = \vert H \vert$$

Therefore $H = H’’$ which is ridiculous. Then what’s wrong with my proof?

Answer 1

It is the fundamental theorem of Galois theory.

If you start with $H\le G$ then $H = Gal(F/F^H)$ and $F/F^H$ is Galois.

For the other subfields $K\subset E\subset F$ with $E\not\supset F^G$ then $Gal(F/E) \subset G $ thus $F^{Gal(F/E)} \supset F^G$, but it is still true that $Gal(F/F^{Gal(F/E)}) = Gal(F/E)$

(the proof is that $Gal(F/E)=Gal(F/F^GE)$ where $F/F^GE$ is Galois)