On normalizers of maximal subfields of division algebras

Document Type : Research Paper

Author

Faculty of Basic Sciences‎, ‎Babol Noshirvani University of Technology‎, ‎Shariati Ave.‎, ‎Babol‎, ‎Post Code 47148-71167‎, ‎Iran.

Abstract

‎Here‎, ‎we investigate a conjecture posed by Amiri and Ariannejad claiming‎ ‎that if every maximal subfield of a division ring $D$ has trivial normalizer‎, ‎then $D$ is commutative‎. ‎Using Amitsur classification of‎ ‎finite subgroups of division rings‎, ‎it is essentially shown that if‎ ‎$D$ is finite dimensional over its center then it contains a maximal‎ ‎subfield with non-trivial normalizer if and only if $D^*$ contains a‎ ‎non-abelian soluble subgroup‎. ‎This result generalizes a theorem of‎ ‎Mahdavi-Hezavehi and Tignol about cyclicity of division algebras of prime index.

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References

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History

• Receive Date: 18 September 2015
• Revise Date: 21 December 2016
• Accept Date: 22 December 2016