Exercise 3, § 7

(a) Let G be a group and M an infinite subset of G. Show that if $G'$ is the subgroup of G generated by M, then $\mathrm{Card}(G') = \mathrm{Card}(M)$.

(b) Let K be a field and M an infinite subset of K. Show that if $K'$ is the subfield of K generated by M, then $\mathrm{Card}(K') = \mathrm{Card}(M)$. (Consider two sequences $(A_n)_{n \geq 0}$, $(P_n)_{n \geq 0}$ of subsets of K such that $A_0 = M$, $P_n$ is the multiplicative subgroup of $K^*$ generated by $A_n \cap K^*$ and $A_{n+1}$ the additive subgroup of K generated by $P_n$; then apply (a).)

(c) Let K be a field and I an infinite set; show that $\mathrm{Card}(K) \leq \dim(K_s^I)$. (Reduce it to the case $I = \mathbf{N}$ and argue by reductio ad absurdum. Let B be a basis of $K_s^\mathbf{N}$ such that $\mathrm{Card}(B) < \mathrm{Card}(K)$ and let L be the subfield of K generated by the coordinates of all the elements of B; then $\mathrm{Card}(L) < \mathrm{Card}(K)$. Form a sequence $(\xi_n)_{n \geq 1}$ of elements of K such that for all n, $\xi_n$ does not belong to the subfield of K generated by L and the $\xi_i$ of index $i < n$; then apply Exercise 2 to obtain a contradiction, by considering the point $x = (\xi_n)$ of $K_s^\mathbf{N}$.)

(d) Deduce from (c) that for every field K and every infinite set I,

("Erdös–Kaplansky theorem").