Non-cyclic algebras with 𝑛-central elements

Matzri, Eliyah; Rowen, Louis; Vishne, Uzi

doi:10.1090/S0002-9939-2011-11236-X

Non-cyclic algebras with $n$-central elements
HTML articles powered by AMS MathViewer

by Eliyah Matzri, Louis H. Rowen and Uzi Vishne PDF

Proc. Amer. Math. Soc. 140 (2012), 513-518 Request permission

Abstract:

We construct, for any prime $p$, a non-cyclic central simple algebra of degree $p^2$ with $p^2$-central elements. This construction answers a problem of Peter Roquette.

References

A. A. Albert, Non-cyclic algebras with pure maximal subfields, Bull. Amer. Math. Soc. 44 (1938), no. 8, 576–579. MR 1563796, DOI 10.1090/S0002-9904-1938-06814-0
A.A. Albert, Modern Higher Algebra, University of Chicago Press, 1937.
A. Adrian Albert, Structure of algebras, American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. Revised printing. MR 0123587
S. A. Amitsur and D. Saltman, Generic Abelian crossed products and $p$-algebras, J. Algebra 51 (1978), no. 1, 76–87. MR 491789, DOI 10.1016/0021-8693(78)90136-9
Roy Dubisch, Non-cyclic algebras of degree four and exponent two with pure maximal subfields, Bull. Amer. Math. Soc. 47 (1941), 131–133. MR 3623, DOI 10.1090/S0002-9904-1941-07389-1
A. Auel, E. Brussel, S. Garibaldi and U. Vishne, Open problems on central simple algebras, Transformation Groups 16(1), 219–264 (2011).
J. Mináč and A. Wadsworth, Division algebras of prime degree and maximal Galois $p$-extensions, Canad. J. Math. 59 (2007), no. 3, 658–672. MR 2319163, DOI 10.4153/CJM-2007-028-7
Uzi Vishne, Galois cohomology of fields without roots of unity, J. Algebra 279 (2004), no. 2, 451–492. MR 2078127, DOI 10.1016/j.jalgebra.2003.12.020