# Are the irreducible rational polynomials of given degree dense?

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 $$\mathbb{Z}$$ ?

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

## 1 Answer

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 $$\sum_{k=0}^n |f_k-g_k| <\varepsilon$$