I asked this question in a similar form on math.se  [here](http://math.stackexchange.com/questions/288297/towers-of-perfect-fields-of-mostly-order-pt), where it has been unanswered for a little over a week some work on it can be found there. The motivation for this question was another [question](http://math.stackexchange.com/questions/286655/number-fields-with-all-degrees-equal-to-a-power-of-three) asked on math.se. 

Fix a prime $p$ and $L$ a number field. Let $f(t) \in L[t]$ be irreducible such that $\deg f(t)=p^{j}$ and $L \subset K$ a field extension of degree $p^k$. What can we say about how $f(t)$ factors over $K[t]$? In particular is it true that $f(t)$ takes a root or has an irreducible factor of order $p^n$ for some $n$?