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August  2019, 13(3): 513-516. doi: 10.3934/amc.2019032

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

Received  September 2018 Revised  November 2018 Published  April 2019

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.

Keywords: Central division algebra, unitary involution, real splitting, positive involution, algebraic coding, unitary space-time code.
Mathematics Subject Classification: Primary: 12E15; Secondary: 11T71, 16W10, 13J30.
Citation: Vincent Astier, Thomas Unger. Galois extensions, positive involutions and an application to unitary space-time coding. Advances in Mathematics of Communications, 2019, 13 (3) : 513-516. doi: 10.3934/amc.2019032
References:
[1]

V. Astier and T. Unger, Positive cones on algebras with involution, preprint, arXiv: 1609.06601.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

V. Astier and T. Unger, Positive cones on algebras with involution, preprint, arXiv: 1609.06601.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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