Exercise 8, § 19

Let $K_0$ be a commutative field of characteristic different from 2, and let $(X_n)_{n\geqslant 1}$ and $(Y_n)_{n\geqslant 1}$ be two infinite sequences of variables and K be the field of rational fractions $K_0((X_n),(Y_n))$. For any $n\geqslant 1$, denote by $F_n$ the field of quaternions over K of type $(X_n,0,Y_n)$.

a) Prove that the tensor product $F =\bigotimes_nF_n$ (III, §4, No. 5, p. 470) is a field with center K such that every subfield generated by a finite subset has finite rank over K (use Exercise 7, a)).

b) Prove that if $u$ is an inner automorphism of F, then there exists a finite subset J of $\mathbf{N}$ such that the restriction of $u$ to $F_n$, for $n \notin J$, is the identity. Deduce that there exists an uncountable set of noninner automorphisms of F.