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A conjecture on permutation trinomials over finite fields of characteristic two
Galois extensions, positive involutions and an application to unitary space-time coding
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland |
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.
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. |
show all references
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. |
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