# Research: Number Theory

## Extensions of the $p$-adic field $\mathbb{Q}_p$ with Galois group $E_1$

Let $p$ be an odd prime number. The graph shows the subgroup lattice of the group $E_1$, which is the unique non-abelian group of order $p^3$ of exponent $p$. $C_p$ denotes the cyclic group with $p$ elements.

The subfield lattice of the unique extension of $\mathbb{Q}_p$ with Galois group $E_1$ with the minimal polynomials for the generating elements of the ramified (sub)extension of degree $p$. Furthermore inertia degrees $(f=p)$ and ramification indices $(e=p,~E=p)$ are given. A proof can be found in the thesis Efficient Enumeration of Extensions of Local Fields with Bounded Discriminant by Sebastian Pauli