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Irreducibility of separable polynomials over the rationals

Let $f(x)\in \mathbb{Q}[x]$ be a polynomial of degree $n$. Suppose $f$ is separable in $\mathbb{R}$.
How to prove that $f$ is irreducible over $\mathbb{Q}$?
It’s clear that the reduction to $\mathbb{R}$ is a finite process, for which we have the property that it takes finitely many steps to “decompose” a separable polynomial over $\mathbb{R}$, i.e. to write it as a sum of pairwise irreducible polynomials. Thus, for a separable polynomial, we have the property that it is irreducible over $\mathbb{R}$ (if it wasn’t already), and hence over $\mathbb{Q}$.

A:

One has to be careful with the word “irreducible”. Some people don’t like this word to be used for non-algebraic (geometric) objects, like polynomials over $Q$. Here is one definition:

A polynomial $f \in \mathbb Q[x_1, \dots, x_n]$ is called irreducible if for any decomposition $f = g \cdot h$ as a product of two non-constant polynomials $g,h \in \mathbb Q[x_1, \dots, x_n]$, one has $deg(g) = deg(h) = 1$.

Obviously, a polynomial is irreducible if and only if it is separable and not a multiple of a linear polynomial. Note that separable polynomials over $\mathbb R$ may be reducible over $\mathbb Q$ in general. So one can’t claim that such polynomials are always irreducible over $\mathbb Q$.
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