Galois extensions, positive involutions and an application to unitary space-time coding

Vincent Astier; Thomas Unger

doi:10.3934/amc.2019032

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2019, Volume 13, Issue 3: 513-516. Doi: 10.3934/amc.2019032

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Galois extensions, positive involutions and an application to unitary space-time coding

Vincent Astier and
Thomas Unger^,

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

^* Corresponding author: Thomas Unger
^* Corresponding author: Thomas Unger

Received: August 31, 2018

Revised: October 31, 2018

Published: August 2019

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Abstract

We show that under certain conditions every maximal symmetric subfield of a central division algebra with positive unitary involution $ (B, \tau) $ will be a Galois extension of the fixed field of $ \tau $ and will "real split" $ (B, \tau) $. As an application we show that a sufficient condition for the existence of positive involutions on certain crossed product division algebras over number fields, considered by Berhuy in the context of unitary space-time coding, is also necessary, proving that Berhuy's construction is optimal.

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Mathematics Subject Classification: Primary: 12E15; Secondary: 11T71, 16W10, 13J30.

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References

[1]	V. Astier and T. Unger, Positive cones on algebras with involution, preprint, arXiv: 1609.06601.
[2]	V. Astier and T. Unger, Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher, J. Algebra, 508 (2018), 339-363. doi: 10.1016/j.jalgebra.2018.05.004.
[3]	G. Berhuy, Algebraic space-time codes based on division algebras with a unitary involution, Adv. Math. Commun., 8 (2014), 167-189. doi: 10.3934/amc.2014.8.167.
[4]	G. Berhuy and F. Oggier, An Introduction to Central Simple Algebras and Their Applications to Wireless Communication, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/191.