# Factor-square polynomials with real coefficients

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?

• Explicit example in terms of coefficients is $f(x)=x^6 - \sqrt{2}x^5 + x^4 - x^2 + \sqrt{2}x - 1$. You can verify that $$f(x^2)=f(x)(x^6 + \sqrt{2}x^5 + (1-\sqrt{2})x^4 - 2x^3 + (1-\sqrt{2})x^2 + \sqrt{2}x + 1).$$ – Sil Apr 25 at 0:54

First notice that if $$\alpha$$ is a root of $$f$$, then $$\alpha^2$$ is also a root of $$f$$ and, since $$f$$ has a finite number of roots, we should have $$\alpha^{2^i}=\alpha^{2^j}$$, thus $$\alpha=0$$ or it is a root of unity. This helps us narrow the search. For an example, let $$\omega$$ be a primitive root of unity with $$\omega^{2^n}=1$$. Then consider $$f(X)=(X-1)^2\prod_{i=1}^{n-1} (X-\omega^{2^i})(X-\overline{\omega}^{2^i}).$$
This obviously has real coefficients, but it will not have integer coefficients. Assuming the contrary, note that it has $$\omega^2$$ as a root which is a primitive root of order $$2^{n-1}$$. Since the cyclotomic polynomial $$\Phi_{2^{n-1}}(X)$$ is the minimal polynomial of $$\omega^2$$ over $$\mathbb{Z}[X]$$, we should have $$\Phi_{2^{n-1}}(X)(X-1)^2|f(x)$$, thus looking at degrees $$2^{n-2}=\varphi(2^{n-1})\leq 2n-2$$, which is false once $$n\geq 6$$.
• How can we be certain that the f(x) you posed is indeed a factor-square polynomial? Also, why must f(x) include $(x-1)^2$? – koisaucer Feb 25 at 21:22