Exercise 2, § 7

Let K be a field, L a subfield of K, I a set and $(x_i)_{1 \leq i \leq m}$ a free family of vectors in $K_s^I$ such that the coordinates of each of the $x_i$ belong to L. Let V be the vector subspace of $K_s^I$ generated by the $x_i$; show that there exists a finite subset J of I, with m elements, such that for every vector $z = (\zeta_\alpha)_{\alpha \in I}$ of V, all the $\zeta_\alpha$ belong to the subfield of K generated by L and the m elements $\zeta_\beta$ where $\beta \in J$. (By applying Corollary 3 of no. 5, to the space $L_s^I$, show that it can be reduced to the case where there exist m indices $\beta_i \in I$ ($1 \leq i \leq m$) such that $\mathrm{pr}_{\beta_i}(x_j) = \delta_{ij}$).