Fields whose torsion free parts divisible with trivial Brauer group

Let $F_0$ be an absolutely algebraic field of characteristic $p>0$ and $\kappa$ an infinite cardinal. It is shown that there exists a field $F$ such that $F^*\cong F^*_0\oplus(\oplus_\kappa \mathbb{Q})$ with $Br(F)=\{0\}$. Let $L$ be an algebraic closure of $F$. Then for any finite subextension $K$ of $L/F$, we have $K^*\cong T(K^*)\oplus(\oplus_\kappa \mathbb{Q})$, where $T(K^*)$ is the group of torsion elements of $K^*$. In addition, $Br(K)=\{0\}$ and $[K:F]=[T(K^*) \cup \{0\}:F_0]$.