Let $F$ be an arbitrary field, $f$ and $g$ irreducible polynomials over $F$. Consider the fields $F[x]/(f(x))$ and $F[y]/(g(y))$. Is there an algorithm to check whether these extensions are isomorphic?
Obviously the polynomials are to have the same degree. Maybe it is somehow possible to "naturally" embed both fields into a larger one and find the basis of their compositum or intersection using something similar to this?