I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois theory course). I have developed and become familiar with the basic properties of valuations, local rings, and the topology/analysis/algebra of the p-adic rings & fields. I am considering pursuing this field further in graduate studies, but I've had no luck searching for open problem in this field, so I'm unsure if this is a useful field to pursue.

For research I am using the books by Koblitz and Gouvea, and then Local Fields by Cassels. I have not seen any mention of any specific points of modern research, other than a mention that $\mathbb{C}_p$ is much less well understood than $\mathbb{C}$ so I am wondering if someone here can help me out.

Thanks a lot. -Robbie