Galois classified the transitive solvable groups of prime degree $p$ (subgroups of the symmetric group ${\frak S}_p$ which are solvable and act transitively on the $p$ letters) .  This is a crucial ingredient in the classification of all separable degree-$p$ extensions of local fields of residual charactertistic $p$.  As an application, one gets an elementary proof of Serre's mass formula in prime  degree.

Seese *Serre's "formule de masse" in prime degree* [arXiv:1005.2016 [math.NT]][1]

See also [Monatshefte 166 (2012) 1, 73--92.][2]


  [1]: http://arxiv.org/abs/1005.2016
  [2]: http://link.springer.com/article/10.1007/s00605-010-0274-0