Algorithm SAGE- PARI? for splitting polynomials over field extension
I though the answer was easier than what I found, so no real chance to do this "by hand". The article from Factoring Polynomials  describes a process over $\mathbb{Q}$ called the Zassenhauss method:  first factor modulo $p$ (say Berlekamp algorithm, rather accessible), then Hensel lift back and check bounds on coefficients and other tests. The Klueners article describes the "van Hoeij" algorithm which I admit is above my math understanding capabilities.

It also describes the "Trager method" over a field extension. Find a primitive element, compute it's minimal polynomial over $\mathbb{Q}$, factor it with say the previous method, so the extension is the field modulo the maximal ideal generated by the minimal polynomial, and apply chinese remainder to decompose the given polynomial.