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

Abstract: For a given unramified extension $K/K_1$ of finite extensions of $\mathbb{Q}_p$, we effectively determine the number of finite Galois extensions $L/K_1$ with maximal unramified subextension $K/K_1$ and a single wild ramification jump at $2$. In fact, we determine explicit formulas in the cases when $K_1/\mathbb{Q}_p$ 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 $k/k_1$ of residue fields.