If you take the splitting field of $x^5+ax+b$ and consider it as an extension of its quadratic subfield, then it will be unramified with Galois group contained in $A_5$ whenever $4a$ and $5b$ are relatively prime. This is a result of YamamotoYamamoto. For almost all $a$ and $b$ (specifically, on the complement of a thin setthin set), the group is $A_5$.
You might also enjoy this preprintpreprint of Kedlaya, which I found very readable. A note on Kedlaya's webpage, dated May 2003, says that he will not be publishing this because it has been superseded by a recent result of Ellenberg and Venkatesh. I assume he is referring to this paperthis paper, but I can't figure out why that one supersedes his.