Fields whose torsion free parts divisible with trivial Brauer group

Year 2022, , 217 - 227, 16.07.2022

Reza Fallah-moghaddam

https://doi.org/10.24330/ieja.1144156

Abstract

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 $[K:F]=[T(K^*) \cup \{0\}:F_0]$.

Keywords

Brauer group, multiplicative group, field, divisibility

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