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Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?

Consider a degree four polynomial $$ f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x] $$ with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...