Example of Galois extension whose Galois group is $A_4$ Can anyone give me an example of a finite Galois extension whose Galois group  is $A_4$?
 A: lhf's links in the comments give good examples over the base field of $\mathbb{Q}$.  If you're willing to accept a slightly larger base field (your question doesn't specify one), you can get a whole slew of examples with very little work.  
Namely, take a random quartic polynomial $f$ with, say, discriminant $\Delta$.  With very high probability (actually, essentially 100%, in a sense that can be made precise), its splitting field $K$ has Galois group $S_4$ over $\mathbb{Q}$.  Now since $\sqrt{\Delta}$ is the product of the differences of the roots of $f$, it lives in $K$, and so whenever $\Delta$ is not a perfect square, $\mathbb{Q}(\sqrt{\Delta})$ is a quadratic extension of $\mathbb{Q}$ contained in $K$.  But now $\operatorname{Gal}(K/\mathbb{Q}(\sqrt{\Delta}))$ is an index-2 subgroup of $\operatorname{Gal}(K/\mathbb{Q})\cong S_4$, the unique example of which is $A_4$.  
So we conclude that $\operatorname{Gal}(K/\mathbb{Q}(\sqrt{\Delta}))\cong A_4$ for almost every quartic polynomial with non-square discriminant.
