Question about existence of Galois extension I'm confused about the distinction between these two:
For a field $F$ and a finite group $G$, we can find a galois extension $E/F$ with galois group $G$. 
Given a finite group $G$, there exists a field $F$ and a galois extension $E/F$ with galois group $G$.
What's the distinction between these statements and are they both true?
 A: The first statement is a generalized statement of the Inverse Galois problem.  For this, the field $F$ is fixed and cannot be chosen.  The second statement removes the restriction: $F$ can be any field.  
If we set $F = \mathbb{Q}$, the first statement is unknown save for a few choices of $G$.  On the other hand, this statement fails when $F$ is a finite field because every Galois extension will have a cyclic Galois group: only $\mathbb{Z}_n$ will appear as a Galois group (for any $n \in \mathbb{N})$.
However, the second statement is true for any finite group $G$.  To see this, first recall Cayley's theorem, which states that any finite group $G$ appears as a subgroup of $S_n$ for $n \geq |G|$.  Next, Hilbert proved that $S_n$ appears as a Galois group over $\mathbb{Q}$; let $K$ be a Galois extension with $\text{Gal}(K/\mathbb{Q}) \cong S_n$.  Finally, the Galois correspondence tells us that, for each subgroup $H$ of $S_n$, there will exist an intermediate field $\mathbb{Q} \subset E_H \subset K$ such that the subgroup is isomorphic to $\text{Gal}(K/E_H)$.  This is to say, $G$ will appear as the Galois group for the extension $K/E_G$.
