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I asked this question in a similar form on math.se herehere, 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 questionquestion 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 $n$?

I asked this question in a similar form on math.se here, 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 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 $n$?

I asked this question in a similar form on math.se here, 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 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 $n$?

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How does an irreducible polynomial of prime power order split over an extension of prime power degree

I asked this question in a similar form on math.se here, 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 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 $n$?