Geodesic
Division algebras of prime degree with infinite genus
Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 264-267.

Voir la notice de l'article provenant de la source Math-Net.Ru

The genus $\mathbf {gen}(\mathcal D)$ of a finite-dimensional central division algebra $\mathcal D$ over a field $F$ is defined as the collection of classes $[\mathcal D']\in \mathrm {Br}(F)$, where $\mathcal D'$ is a central division $F$-algebra having the same maximal subfields as $\mathcal D$. For any prime $p$, we construct a division algebra of degree $p$ with infinite genus. Moreover, we show that there exists a field $K$ such that there are infinitely many nonisomorphic central division $K$-algebras of degree $p$ and any two such algebras have the same genus. 
@article{TRSPY_2016_292_a15,
     author = {S. V. Tikhonov},
     title = {Division algebras of prime degree with infinite genus},
     journal = {Informatics and Automation},
     pages = {264--267},
     publisher = {mathdoc},
     volume = {292},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a15/}
}
TY  - JOUR
AU  - S. V. Tikhonov
TI  - Division algebras of prime degree with infinite genus
JO  - Informatics and Automation
PY  - 2016
SP  - 264
EP  - 267
VL  - 292
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a15/
LA  - ru
ID  - TRSPY_2016_292_a15
ER  - 
%0 Journal Article
%A S. V. Tikhonov
%T Division algebras of prime degree with infinite genus
%J Informatics and Automation
%D 2016
%P 264-267
%V 292
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a15/
%G ru
%F TRSPY_2016_292_a15
S. V. Tikhonov. Division algebras of prime degree with infinite genus. Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 264-267. http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a15/

[1] Amitsur S.A., “Generic splitting fields of central simple algebras”, Ann. Math. Ser. 2, 62 (1955), 8–43 | DOI | MR | Zbl

[2] Chernousov V.I., Rapinchuk A.S., Rapinchuk I.A., “On the genus of a division algebra”, C. r. Math. Acad. sci. Paris, 350:17–18 (2012), 807–812 | DOI | MR | Zbl

[3] Chernousov V.I., Rapinchuk A.S., Rapinchuk I.A., “The genus of a division algebra and the unramified Brauer group”, Bull. Math. Sci., 3:2 (2013), 211–240 | DOI | MR | Zbl

[4] Garibaldi S., Saltman D.J., “Quaternion algebras with the same subfields”, Quadratic forms, linear algebraic groups, and cohomology, Ed. by J.-L. Colliot-Thélène et al., Springer, New York, 2010, 225–238 | DOI | MR | Zbl

[5] Krashen D., McKinnie K., “Distinguishing division algebras by finite splitting fields”, Manuscr. math., 134:1–2 (2011), 171–182 | DOI | MR | Zbl

[6] Meyer J.S., “Division algebras with infinite genus”, Bull. London Math. Soc., 46:3 (2014), 463–468 | DOI | MR | Zbl

[7] Rapinchuk A.S., Rapinchuk I.A., “On division algebras having the same maximal subfields”, Manuscr. math., 132:3–4 (2010), 273–293 | DOI | MR | Zbl

[8] Rehmann U., Tikhonov S.V., Yanchevskiĭ V.I., “Prescribed behavior of central simple algebras after scalar extension”, J. Algebra, 351:1 (2012), 279–293 | DOI | MR | Zbl

[9] Saltman D.J., Lectures on division algebras, Amer. Math. Soc., Providence, RI, 1999 | MR | Zbl