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 [Yamamoto][1]. For almost all $a$ and $b$ (specifically, on the complement of a [thin set][2]), the group is $A_5$.

You might also enjoy this [preprint][3] 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 A note on Kedlaya's webpage, dated May 2003, says that he will not be publishing this because it has been superseded by a recent result of Ellenberg and Venkatesh.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 paper][4], but I can't figure out why that one supersedes his.


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=266898
  [2]: https://en.wikipedia.org/wiki/Thin_set_%28Serre%29
  [3]: https://web.archive.org/web/20081031175638/http://math.mit.edu/~kedlaya/papers/unramified.ps.gz
  [4]: https://arxiv.org/abs/math.NT/0309153