My earlier related question Lower Degree Elements in an Algebraic Number Field has been given a clean answer for the first part. My present question is below:

Take a number field $K=\mathbf{Q}(\alpha)$ of degree $n$ admitting intermediate fields, so $n$ is a composite number $n=mk$, and there are algebraic numbers of degree $m$ inside $K$. Is there a polynomial $f(x)\in \mathbf{Q}[x]$ of degree $k$ such that $f(\alpha)$ is an algebraic number of degree $m$?

I am looking for a way of reaching lower degree elements from the primitive elements. A 'generic polynomial' evaluates to another primitive element, there must be some special property that will give rise to lower degree elements. `Complementary degree' does not seem to be enough.

Example: A primitive $p^{a+b}$-th root of unity, a number of degree $p^{a+b-1}(p-1)$, when raised to the power $p^a$, leads to a primitive $p^b$-th root of unity, a number of desired degree I am looking for.