I was busy doing a homework exercise in which I had to compute the discriminant $\Delta(f)$ of the polynomial
$$f(X) = X^4+X^2+X+1$$ which turned out to be the prime $257$. Subsequently, I was asked to show that $f$ is irreducible over $\Bbb Q[X]$ (straightforward to do directly since only linear and quadratic factors need to be considered).
However, we also have the general identity: $g \mid f \implies \Delta(g) \mid \Delta(f)$ so that any proper factorisation of $f$ needs a factor with discriminant $1$. I was trying out a few values and came to the conjecture that the only irreducible monic polynomials in $\Bbb Z[X]$ having discriminant $1$ are the linear monic polynomials (i.e. the trivial examples).
I was however unable to prove or disprove this conjecture, or find any information about it. I'd like to know whether it holds; if possible with proof/counterexample and/or references. Thanks in advance.