I  asked this  [question at MSE](http://math.stackexchange.com/questions/1438481/a-question-on-galois-theory) but I did not receive an answer. So I ask it at MO:

We denote the field of rational numbers by $\mathbb{Q}$. The Galois group of a polynomial $f$ is  denoted by $Gal(f)$. The commutator subgroup of a group $G$ is denoted by $G'$.

>Is there a $\mathbb{Q}$-linear map $T$ over $\mathbb{Q}[x]$ such that for all polynomials $f\in \mathbb{Q}[x]$ we have $$Gal(T(f))=(Gal(f))'$$

For a related post see the following question:

http://mathoverflow.net/questions/219455/endofunctors-on-the-category-of-groups-which-are-galois-related-to-a-linear-map