Geodesic
Abelian crossed products and scalar extensions of central simple algebras
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 17, Tome 356 (2008), pp. 179-188 
S. V. Tikhonov; V. I. Yanchevskii. Abelian crossed products and scalar extensions of central simple algebras. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 17, Tome 356 (2008), pp. 179-188. http://geodesic.mathdoc.fr/item/ZNSL_2008_356_a6/
@article{ZNSL_2008_356_a6,
     author = {S. V. Tikhonov and V. I. Yanchevskii},
     title = {Abelian crossed products and scalar extensions of central simple algebras},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {179--188},
     year = {2008},
     volume = {356},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_356_a6/}
}
TY  - JOUR
AU  - S. V. Tikhonov
AU  - V. I. Yanchevskii
TI  - Abelian crossed products and scalar extensions of central simple algebras
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2008
SP  - 179
EP  - 188
VL  - 356
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2008_356_a6/
LA  - ru
ID  - ZNSL_2008_356_a6
ER  - 
%0 Journal Article
%A S. V. Tikhonov
%A V. I. Yanchevskii
%T Abelian crossed products and scalar extensions of central simple algebras
%J Zapiski Nauchnykh Seminarov POMI
%D 2008
%P 179-188
%V 356
%U http://geodesic.mathdoc.fr/item/ZNSL_2008_356_a6/
%G ru
%F ZNSL_2008_356_a6

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $\mathcal A$ be a central simple algebra over a field $k$ and $G$ a commutative group with $|G|=\deg(\mathcal A)$. We prove that there exists a regular field extension $E/k$ preserving indices of $k$-algebras such that $\mathcal A\otimes_k E$ is a crossed product with the group $G$. Bibl. – 11 titles.

[1] N. Burbaki, Algebra. Mnogochleny i polya. Uporyadochennye gruppy, gl. IV–VI, Nauka, M., 1965 | MR

[2] R. Pirs, Assotsiativnye algebry, Mir, M., 1986 | MR

[3] S. A. Amitsur, “On central division algebras”, Israel J. Math., 12 (1972), 408–420 | DOI | MR | Zbl

[4] Ph. Gille, T. Szamuely, Central Simple Algebras And Galois Cohomology, Cambridge Univ. Press, Cambridge, 2006 | MR

[5] J. Harris, Algebraic geometry. A first course, Graduate Texts in Math., 133, Springer-Verlag, Berlin et al., 1992 | MR | Zbl

[6] N. Jacobson, Finite-dimensional division algebras, Springer-Verlag, Berlin et al., 1996 | MR | Zbl

[7] A. S. Merkurjev, I. A. Panin, A. R. Wadsworth, A of index reduction formulas, Preprint 94-079/ Preprint-Server of the SFB 343 “Diskrete Strukturen in der Mathematik”

[8] H. Miki, “On Grunwald–Hasse–Wang's theorem”, J. Math. Soc. Japan, 30 (1978), 313–325 | DOI | MR | Zbl

[9] B. È. Kunyavskiĭ, L. H. Rowen, S. V. Tikhonov, V. I. Yanchevskiĭ, “Bicyclic algebras of prime exponent over function fields”, Trans. Amer. Math. Soc., 358 (2006), 2579–2610 | DOI | MR | Zbl

[10] U. Rehmann, S. V. Tikhonov, V. I. Yanchevskiĭ, Cyclicity of algebras after a scalar extension, Gotovitsya k pechati

[11] M. Van den Berg, A. Schofield, “The index of a Brauer class on a Brauer–Severi variety”, Trans. Amer. Math. Soc., 333 (1992), 729–739 | DOI | MR | Zbl