A polynomial $f(x)$ is a factor-square polynomial if $f(x)|f(x^2)$. For example, $x-1|x^2-1$, so $x-1$ is a factor-square polynomial.
Here is a list of polynomials degrees 1 and 2: $𝑥$, $𝑥−1$; $x^2$, $x^2-1$, $x^2-x$, $x^2+x+1$, $x^2-2x+1$.
After generating some more of these types of polynomials, it appears that either all of the coefficients are real, or all complex and irreducible over reals.
Are there monic factor-square polynomials of some degree with real number coefficients, but some of those coefficients are not integers?