Let $p\ne 2$ be a prime. We derive a technique based on local class field theory and on the expansions of certain resultants allowing to recover very easily Lbekkouri’s characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over ${\mathbb{Q}}_p$, and extend it to when the base fields $K$ is an unramified extension of ${\mathbb{Q}}_p$.When a polynomial satisfies a subset of such conditions the first unsatisfied condition characterizes the Galois group of the normal closure. We derive a complete classification...

We present an algorithm that returns a proper factor of a polynomial $\Phi \left(x\right)$ over the $p$-adic integers ${\mathbb{Z}}_p$ (if $\Phi \left(x\right)$ is reducible over ${\mathbb{Q}}_p$) or returns a power basis of the ring of integers of ${\mathbb{Q}}_p\left[x\right]/\Phi \left(x\right){\mathbb{Q}}_p\left[x\right]$ (if $\Phi \left(x\right)$ is irreducible over ${\mathbb{Q}}_p$). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.