Reducibility of Cyclotomic polynomials over integers

Question

Cyclotomic Polynomials $\Phi_n(x)$ are irreducible themselves, but when restricted to certain integers (in terms of $x$), may be reducible. The most obvious case is when $x$ is a perfect power (simple algebraic factorizations exist). Another example, $\Phi_6(3x^2)$ = $(3x^2-3x+1)(3x^2+3x+1)$, but $3x^2$ is not a perfect power itself (in general, for all $x$). Another example is $\Phi_5(x^3+x^2+2x)=(x^4 + 2x^3 + 4x^2 + 3x + 1)(x^8 + 2x^7 + 6x^6 + 6x^5 + 9x^4 + 3x^3 + 4x^2 - x + 1)$. Question is, suppose $p(x)$ is a polynomial defining a field $K$. If the $n$-th cyclotomic field and $K$ are isomorphic, is there a reducible cyclotomic polynomial $\Phi_n(q(x))$ which contains $p(x)$ as a factor? For example, the fields defined by $\Phi_3(x) = x^2+x+1$ and $p(x) = x^2+x+7$ are isomorphic, so there should exist a polynomial $q(x)$ such that $x^2+x+7$ divides $\Phi_3(q(x))$. How to find $q(x)$ (in any given case) I am unsure.