Fields whose torsion free parts divisible with trivial Brauer group

Year 2022, Volume: 32 Issue: 32, 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

References

• J.-L. Colliot-Thelene, R. Guralnick and R. Wiegand, Multiplicative groups of fields modulo products of subfields, J. Pure Appl. Algebra, 106(3) (1996), 233-262.
• M. Contessa, J. Mott and W. Nichols, Multiplicative groups of fields, in: Advances in Commutative Ring Theory, Fez 1997, in: Lect. Notes Pure Appl. Math., Vol. 205, Dekker, New York, 197-216, 1999.
• R. M. Dicker, A set of independent axioms for a field and a condition for a group to be the multiplicative group of a field}, Proc. London Math. Soc., 18(3) (1968), 114-124.
• P. K. Draxl, Skew Fields, Cambridge Univ. Press, Cambridge, 1983.
• L. Fuchs, Abelian Groups, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York-Oxford-London-Paris, 1960.
• G. Karpilovsky, Unit Groups of Classical Rings, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988.
• T. Y. Lam, A First Course in Noncommutative Rings, Second edition, Grad. Texts in Math., vol. 131, Springer-Verlag, New York, 2001.
• S. Lang, Algebra, Third edition, Grad. Texts in Math., Vol. 211, Springer-Verlag, New York, 2002.
• W. May, Multiplicative groups of fields, Proc. London Math. Soc., 24 (1972), 295-306.
• W. May, Multiplicative groups under field extensions, Canadian J. Math., 31 (1979), 436-440.
• A.S. Merkurjev, Brauer groups of fields, Comm. Algebra, 11(22) (1983), 2611-2624.
• P. Morandi, Field and Galois Theory, Springer-Verlag, New York, 1996.
• G. Oman, Divisible multiplicative groups of fields, J. Algebra, 453 (2016), 177-188.
• R. S. Pierce, Associative Algebras, Grad. Text in Math., Vol. 88, Springer-Verlag, New York-Berlin, 1982.
• D. J. S. Robinson, A Course in The Theory of Groups, Second edition, Grad. Texts in Math., Vol. 80, Springer-Verlag, New York, 1996.
• W. Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin, 1985.
• G. Whaples, The generality of local class field theory, Proc. Amer. Math. Soc., 8 (1957), 137-140.