Question 48
Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$) such that $f$ is irreducible in $\mathbb{Q}[X]$, and $f(X^2)$ is reducible in $\mathbb{Q}[X]$. Show that $f = \pm(u^2 - Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.
For instance, if $a$ and $b$ are coprime integers and $a$ is odd, then $f = a^4 X^2 + 4b^4$ is a primitive polynomial in $\mathbb{Z}[X]$, irreducible in $\mathbb{R}[X] \supset \mathbb{Q}[X]$, $f(X^2) = a^4 X^4 + 4b^4 = (a^2 X^2 - 2abX + 2b^2)(a^2 X^2 + 2abX + 2b^2)$ is reducible in $\mathbb{Z}[X] \subset \mathbb{Q}[X]$, and $f = (a^2 X + 2b^2)^2 - X \cdot (2ab)^2$.
For instance, if $a$ and $b$ are coprime integers and $a$ is odd, then $f = a^4 X^2 + 4b^4$ is a primitive polynomial in $\mathbb{Z}[X]$, irreducible in $\mathbb{R}[X] \supset \mathbb{Q}[X]$, $f(X^2) = a^4 X^4 + 4b^4 = (a^2 X^2 - 2abX + 2b^2)(a^2 X^2 + 2abX + 2b^2)$ is reducible in $\mathbb{Z}[X] \subset \mathbb{Q}[X]$, and $f = (a^2 X + 2b^2)^2 - X \cdot (2ab)^2$.