On Galois Extensions of Local Fields with a Single Wild Ramification Jump

For a given unramified extension K/K₁ of finite extensions of Qₚ, we effectively determine the number of finite Galois extensions L/K₁ with maximal unramified subextension K/K₁ and a single wild ramification jump at $3$. In fact, we determine explicit formulas in the cases when K₁/Qₚ is totally ramified and when it is unramified. This builds upon the tamely ramified case, which is a classical consequence of Serre's Mass Formula, exhibiting a more restrictive behavior than in the tamely ramified case because the degrees of such extensions are bounded. As a consequence of the calculation, it is also found that the count depends only on the extension k/k₁ of residue fields.