Which finite groups occur as Galois groups of some field extension?
Solution
Every finite group occurs as the Galois group of some field extension.
Let \(G\) be finite and choose a field \(k\). Form the rational function field \[ L=k(x_g:g\in G) \] in algebraically independent variables indexed by \(G\). Let \(G\) act on \(L\) by \[ h(x_g)=x_{hg} \qquad(h,g\in G). \] This action is faithful. Let \[ K=L^G \] be the fixed field. Artin’s theorem on fixed fields of finite groups of automorphisms gives \[ [L:K]=|G| \] and \[ \operatorname{Gal}(L/K)\cong G. \]
Thus there is no restriction if the base field is allowed to vary.
By contrast, asking which finite groups occur as Galois groups over the specific field \(\QQ\) is the inverse Galois problem over \(\QQ\), which is not known in full generality.