I want find the polynomials which satisfy the condition $$f(x)\mid f(x^2).$$
I want to find such polynomials with integer coefficients, real number coefficients and complex number coefficients.
For example, $x$ and $x-1$ are the linear polynomials which satisfy this condition.
Here is one way to find the $2$-degree polynomials with integer coefficients. Let the quadratic be $p=ax^2+bx+c$, so its value at $x^2$ is $q=ax^4+bx^2+c$. If $p$ is to be a divisor of $q$ let the other factor be $dx^2+ex+f.$ Equating coefficients gives equations
Now we know $a,c$ are nonzero (else $p$ is not quadratic, or is reducible). So from  and  we have $d=f=1.$ Then from  and  we obtain $ae=ce.$ Here $e=0$ leads to $b=0$ from either  or , and  then reads $a+c=0$, so that $p=a(x^2-1)$ which is reducible. So we may assume $e$ is nonzero, and also $a=c.$
At this point,  and  say the same thing, namely $ae+b=0.$ So we may replace $b=-ae$ in  (with its $c$ replaced by $a$) obtaining $a+(-ae)e+a=-ae,$ which on factoring gives $a(2-e)(e+1)=0.$ The possibility $e=2$ then leads after some algebra to $2a+b=0$ and $p=a(x-1)^2$ which is reducible, while the possibility $e=-1$ leads to $a=b$ and then $p=ax^2+ax+a$ as claimed.
Should we list out all the irreducible degree polynomials and then check if these polynomials satisfy the condition
- $x^2 + x + 1$
- $x^3 + x^2 + 1$
- $x^3 + x + 1$
- $ x^4 + x^3 + x^2 + x + 1 $
- $ x^4 + x^3 + 1 $
- $ x^4 + x + 1 $
With the real number coefficients which can be factored into
$$(x-c_1)(x-c_2)\cdots(x^2-2a_1x-(a_1^2+b_1^2))(x^2-2a_2x-(a_2^2+b_2^2))\cdots$$ If all of these linear terms and quadratic terms satisfy $$f(x)\mid f(x^2),$$ this polynomial satisfy too? So what's pattern in the real number polynomials?