Fields whose torsion free parts divisible with trivial Brauer group

Authors : Reza FALLAHMOGHADDAM

Pages : 217-227

Doi:10.24330/ieja.1144156

View : 30 | Download : 6

Publication Date : 2022-07-16

Article Type : Research Paper

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 $F^*\cong F^*_0\oplusinsert ignore into journalissuearticles values(\oplus_\kappa \mathbb{Q});$ with $Brinsert ignore into journalissuearticles values(F);=\{0\}$. Let $L$ be an algebraic closure of $F$. Then for any finite subextension $K$ of $L/F$, we have $K^*\cong Tinsert ignore into journalissuearticles values(K^*);\oplusinsert ignore into journalissuearticles values(\oplus_\kappa \mathbb{Q});$, where $Tinsert ignore into journalissuearticles values(K^*);$ is the group of torsion elements of $K^*$. In addition, $Brinsert ignore into journalissuearticles values(K);=\{0\}$ and $[K:F]=[Tinsert ignore into journalissuearticles values(K^*); \cup \{0\}:F_0]$.
Keywords : Brauer group, multiplicative group, field, divisibility

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