# Find the polynomials which satisfy the condition $f(x) \mid f(x^2)$ called the factor square property

The Factor Square Property (FSP) is the divisibility of the polynomial $f(x^2)$ by $f(x)$.

1. Is $x^2+x+1$ the only FSP irreducible polynomial of degree $2$ ?

2. Are there other linear polynomial besides $x$ and $x-1$ with FSP?

3. Do we have other FSP irreducible polynomials of degree $3$ or $4$? Any of these have integer coefficients??

4. Are there any other observations you can make about polynomials with FSP?

So this question has been posted before in: Link 1 Link 2

But the solutions use cyclotomic polynomials. Is there an easier solution? The Question is from Ross MathCamp, so, I suppose they will give some observation based question which doesn't require any kind of Complex Numbers I suppose.

Hint: Assuming that $f(x)$ is a monic and non-constant polynomial, $f(x)\mid f(x^2)$ implies that all the complex roots of $f(x)$ are roots of $f(x^2)$ too, hence that the set of roots of $f(x)$ is closed with respect to squaring. By considering the $n$-th cyclotomic polynomial with $n$ being odd, we always have that $\Phi_n(x)$ is a divisor of $\Phi_n(x^2)$, hence there are polynomials with the FSP with arbitrarily large degree (the degree of $\Phi_n(x)$ is $\varphi(n)$).
• @TreFox: otherwise $\Phi_n(x)\mid \Phi_n(x^2)$ is not granted. For instance $\Phi_4(x)=x^2+1$ is not a divisor of $x^4+1$. – Jack D'Aurizio Jan 9 '18 at 14:47