Independence of $\mathrm{\ell }$-adic Galois representations over function fields

Let $K$ be a finitely generated extension of $\mathbb{Q}$. We consider the family of $\mathrm{\ell }$-adic representations ($\mathrm{\ell }$ varies through the set of all prime numbers) of the absolute Galois group of $K$, attached to $\mathrm{\ell }$-adic cohomology of a smooth separated scheme of finite type over $K$. We prove that the fields cut out from the algebraic closure of $K$ by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question asked by Serre in 1991.