We know that $GF(p^c)$ is a subfield of $GF(p^{cn})$. Also we know that elements in $GF(p^c)$ can be represent by degree $c$ polynomials with coefficients in $\mathbb Z_p$, where multiplication is done by usual polynomial multiplication modulo a degree $c$ irreducible polynomial $p$.

The question is, given the representation of $GF(p^c)$ and $GF(p^{cn})$ (by giving the two irreducible polynomials), can we find what an element in $GF(p^c)$ should be in the representation of $GF(p^{cn})$? Can it be found in polylog($p^c$) time?

It would be really helpful if you could also provide some references.