[Math] factor-square property (FSP) of polynomials

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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?

Best Answer

A google search of "factor square property" led only to this present question. So recall from comment that a polynomial $f(x)$ which divides $f(x^2)$ is said to be FSP or to have the factor square property.

This addresses [1], and shows an irreducible FSP quadratic must be $ax^2+ax+a$, which depending on what "irreducible" means might also imply $a=1$. We also address [3], giving a source of examples.

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

[1] $ad=a,$

[2] $ae+bd=0,$

[3] $af+be+cd=b,$

[4] $bf+ce=0,$

[5] $cf=c.$

Now we know $a,c$ are nonzero (else $p$ is not quadratic, or is reducible). So from [1] and [5] we have $d=f=1.$ Then from [2] and [4] we obtain $ae=ce.$ Here $e=0$ leads to $b=0$ from either [2] or [4], and [3] 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, [2] and [4] say the same thing, namely $ae+b=0.$ So we may replace $b=-ae$ in [3] (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.

For some higher degree examples, for any odd prime $p$ the cyclotomic polynomial $f(x)=(x^p-1)/(x-1)$ is seen to be FSP because $$\frac{f(x^2)}{f(x)}=\frac{x^{2p}-1}{x^2-1}\cdot \frac{x-1}{x^p-1} \\ =\frac{x^p+1}{x+1}.$$ For $p=5$ this is the polynomial $$x^4+x^3+x^2+x+1.$$ Note that for odd prime $p$ these cyclotomic polynomials are known to be irreducible (there is a simple Eisenstein proof using a change of variables).

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