If $f\in \mathbb{Q}[X]$ is a polynomial of degree $n\geq 2$ and $\varepsilon>0$, is there always an irreducible $g\in \mathbb{Q}[X]$ of degree $n$ with $$\sum_{k=0}^n |f_k-g_k| <\varepsilon\quad\text{ (where $f_k$, $g_k$ are the coefficients of $f,g$)}$$ ?

edit : My idea so far: Change the coefficients so that the denominators are large integers, which leaves more room for changing the numerators and apply a irreducibility criterion for polynomials in $\mathbb{Z}$ ?

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    $\begingroup$ My first thoughts on this is to cleverly use the Eisenstein criterion. $\endgroup$ – Jakobian Jan 5 at 19:52

Following @Jakobian's comment:

For $f\in\mathbb{Q}[X]$ choose $N\in\mathbb{N}$ such that $N\cdot f\in\mathbb{Z}[X]$. For large $l\in\mathbb{N}$ consider the polynomials $F:=2^l N f$ and $G$ with $G_k:=F_k; 1\leq k\leq n-1$; $G_0=F_0+2$; $G_n=F_n+1$. By Eisenstein criterion $G$ and $g:=\frac{G}{2^l}$ are irreducible. For suitable large $l$ we get $$\sum_{k=0}^n |f_k-g_k| <\varepsilon$$


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