TITLE: Why these 'elementary' facts do not solve the Inverse Galois Problem?
QUESTION [15 upvotes]: Since every finite group $G$ is isomorphic to a subgroup of $S_{n}$ and according to the first answer on this question there is always (for all $n\geq 1$) a finite Galois extension $K/\mathbb{Q}$ with $\operatorname{Gal}(K/\mathbb{Q})\cong S_{n}$, doesn't the Fundamental Theorem of Galois Theory gives a positive answer to the Inverse Galois Problem?
Where is the obvious point I am missing? Thanks!

REPLY [19 votes]: If $H\subset \mathrm{Gal}(K/\mathbb Q)$ then the set elements elements of $K$ fixed by $H$ is a field $k$, and you get $$H\cong\mathrm{Gal}(K/k)\,.$$ 
But the inverse Galois question is seeking $k$ so that $H\cong \mathrm{Gal}(k/\mathbb Q)$.