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]

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

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.

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

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