Suppose that over some finite field $GF(q)$, we have two monic primitive polynomials of orders $n$ and $mn$.
-From these polynomials, is there always a 'natural' monic primitive polynomial over $GF(q^{n})$ of order $m$?
-Is there an algorithm for finding this polynomial directly? (i.e. one which doesn't rely on testing ~$q^{mn}$ polynomials)