I asked this question in a similar form on math.se [here](https://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](https://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 $n$? 



How does an irreducible polynomial of prime power order split over an extension of prime power degree
Just for concreteness, here's an explicit counterexample along the lines of my comment. Let $L=\mathbb{Q}$ and let $f(t)=t^7 - 7t+3$ and $g(t)=t^7-14t^4+42t^2-21t-9$. Then $f(t)$ and $g(t)$ are irreducible in $L[t]$. Now let $K=\mathbb{Q}(c)$ where $g(c)=0$, so $[K:L]=7$. Then $f(t)$ factors in $K[t]$ into irreducibles of degrees $3$ and $4$. For instance, this can be verified using the following Magma code, which can be used on the [free Magma online calculator](http://magma.maths.usyd.edu.au/calc):



```
_<t>:=PolynomialRing(Rationals());
f:=t^7-7*t+3;
g:=t^7-14*t^4+42*t^2-21*t-9;
IsIrreducible(f);
K<c>:=NumberField(g);
_<x>:=PolynomialRing(K);
Factorization(Evaluate(f,x));

```

